1

Residual stresses in cold-rolled stainless steel hollow sections

M. Jandera1, L. Gardner2, J. Machacek3

1 PhD student, corresponding author, Faculty of Civil Engineering, Czech Technical

University in Prague, Thakurova 7, Praha, 166 29, Czech Republic.

Email: michal.jandera@fsv.cvut.cz

2 Senior Lecturer in Structural Engineering, Department of Civil and Environmental

Engineering, Imperial College London, SW7 2AZ, United Kingdom. Email:

leroy.gardner@imperial.ac.uk

3 Professor, Faculty of Civil Engineering, Czech Technical University in Prague,

Thakurova 7, Praha, 166 29, Czech Republic. Email: machacek@fsv.cvut.cz

Abstract

Stainless steel exhibits a pronounced response to cold-work and heat input. As a result, the

behaviour of structural stainless steel sections, as influenced by strength, ductility and residual

stress presence, is sensitive to the precise means by which the sections are produced. This

paper explores the presence and influence of residual stresses in cold-rolled stainless steel box

sections using experimental and numerical techniques. In previous studies, residual stress

magnitudes have been inferred from surface strain measurements and an assumed through-

thickness stress distribution. In the present study, through-thickness residual stresses in cold-

rolled stainless steel box sections have been measured directly by means of X-ray diffraction

and their effect on structural behaviour has been carefully assessed through detailed non-

linear numerical modelling. Geometric imperfections, flat and corner material properties and

the average compressive response of stainless steel box sections were also examined

experimentally and the results have been fully reported. From the X-ray diffraction

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measurements, it was concluded that the influence of through-thickness (bending) residual

stresses in cold-rolled stainless steel box sections could be effectively represented by a

rectangular stress block distribution. The developed ABAQUS numerical models included

features such as non-linear material stress-strain characteristics, initial geometric

imperfections, residual stresses (membrane and bending) and enhanced strength corner

properties. The residual stresses, together with the corresponding plastic strains, were

included in the FE models by means of the SIGINI and HARDINI Fortran subroutines. Of the

two residual stress components, the bending residual stresses were found to be larger in

magnitude and of greater (often positive) influence on the structural behaviour of thin-walled

cold-formed stainless steel sections.

Keywords: Buckling, Experiments, Hollow section, Local buckling, Residual stress, Stainless

steel, Stub column, Structural testing, X-ray diffraction

1. Introduction

Recent years have seen an increasing demand for highly alloyed steels, including high

strength and stainless steels, in structural applications. Stainless steel is gaining more

widespread usage as a load bearing construction material due to its combination of corrosion

resistance, attractive surface finish and structural qualities. The present research concerns

cold-rolled stainless steel box sections, which are suitable and widely used as compression

elements in structures.

Stainless steel exhibits dissimilar characteristics to carbon steel, such as non-linear and

asymmetric stress-strain behaviour, anisotropy, pronounced response to cold-working

processes, different influences of initial imperfections and different thermal properties [1, 2,

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3]. Therefore, residual stress patterns and their influence on stainless steel structural elements

cannot simply be based on carbon steel research. Over the past few decades, a significant

volume of research world-wide has been devoted to the study of residual stresses in structural

elements. This has focused mainly on welded and hot-rolled carbon steel profiles (e.g. [4, 5] )

and more recently on cold-formed carbon steel sections as well [6]. For stainless steel, the

most comprehensive measurements were made by Cruise and Gardner [7] on hot-rolled,

press-braked and cold-formed sections. The highest magnitudes of residual stresses were

found in cold-rolled box sections. These sections also form the basis of the present study

where the compressive response and influence of residual stresses are examined. Other

residual stress measurements on a high strength austenitic stainless steel rectangular hollow

section (RHS 200×110×4) were presented by Young and Lui [8].

2. Experimental investigation

The experimental programme carried out in this study comprised X-ray diffraction

measurements of residual stresses in a stainless steel rectangular hollow section (RHS) and

stub column tests of square hollow sections (SHS), the results of which were subsequently

used for FE model verification. All tested material was Grade 1.4301 stainless steel, since this

is commonly used in structural applications. Accompanying measurements included material

tests on the flat and corner areas of the SHS as well as initial geometric imperfection studies.

The authors are aware of no previous measurements of this kind on structural stainless steel

sections.

2.1 Residual stress measurements by X-ray diffraction

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Residual stresses in structural sections are most commonly assessed by the sectioning

technique, in which residual stress magnitudes are inferred from a combination of measured

surface strains and an assumed through-thickness stress distribution. X-ray diffraction enables

the direct evaluation of residual stresses through the material thickness. The X-ray diffraction

method is based on measuring changes of crystalline plane distances in material

microstructures, revealing elastic strains as described by Brag's law [9]. Surface stresses may

be obtained up to depths of 5-10 μm. For through-thickness measurements, separate layers

have to be electrolytically removed. In the described measurements, a 1.8 mm diameter X-ray

beam was used together with a 10 mm specimen oscillation along the specimen’s axis to

improve the reliability of the measurements. The X-ray diffraction measurement apparatus

and configuration are shown in Fig 1.

A total of 20 surface measurements and two half through-thickness measurements in the weld

area of a RHS 100×80×2 specimen were successfully performed in both the longitudinal and

transverse directions, with respect to the rolling direction. Eight further through-thickness

measurements were performed but these yielded unreliable diffraction patterns due to the

large material grain size found in the austenitic microstructure (probably exceeding 100 μm,

[9]). The surface measurements (Fig. 2) revealed a thin layer of compressive residual stress on

the outer faces of the specimen, which is believed to have been induced by contact with the

forming tools [10] and can be beneficial terms of enhancing stress-corrosion-cracking

resistance. The magnitudes of these surface stresses in the transverse direction were found to

be, on average, about two times those in the longitudinal direction.

The two through-thickness measurements taken at point 3 (the weld location) and at point 1 (8

mm from the weld) – see Fig. 3 – generally revealed tensile longitudinal residual stresses

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through the outer half-thickness of the specimen, with the exception of the thin compressive

surface layer described above. These stresses appear to be approximately uniform through the

half-thickness, and may therefore be represented as a rectangular stress block distribution, as

assumed by Cruise [11] and Cruise and Gardner [7] when establishing residual stress

magnitudes from released surface strains following sectioning. This hypothesis does, however

require further verification through additional measurements.

2.2 Material properties

The high residual stresses reported in the previous sub-section are indicative of large plastic

straining during production; in this sub-section, the associated strength enhancements due to

cold-work are assessed through tensile material testing. Tensile coupons were extracted from

the flat (F) and corner (C) regions of each of the tested section sizes. The key material

parameters have been summarised in Table 1, where E is the initial tangent (Young’s)

modulus, 0.2 and 1.0 are the 0.2% and 1.0% proof strengths respectively, u is the ultimate

tensile strength and n and n’0.2,1.0 are strain hardening exponents for the compound Ramberg-

Osgood material model described in [12]. The results of the corner coupon tests were

compared with the simple predictive relationship observed in [13, 14], as given by Eq. (1).

f,uc,2.0 85.0 (1)

where σ0.2,c is the 0.2 % proof strength of material in the corner area and σu,f is the ultimate

strength of the flat part of the section.

The expression (Eq. (1)) was subsequently refined by Ashraf et al. [15] and re-evaluated

recently [16]. Fig. 4 illustrates good correspondence between the measured data from the

present study and Eq. (1). A proportionality constant of 0.83 is indicated by the test results

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presented herein in comparison to the originally proposed 0.85, confirming the validity of the

original model.

2.3 Initial geometric imperfections

Initial geometric imperfections were measured for all tested specimens in a grid of points, as

indicated in Fig. 5, using 0.01 mm precision dial gauge. The imperfections were measured

relative to the straight edges of the sections. The largest imperfections were generally

observed near to the ends of each specimens; this was attributed to the relaxed bending

residual stresses. In numerical studies, use of measured (true) initial geometric imperfections

is rare; typically elastic eigenmode shapes are employed instead. This approach was also

followed in this study. The maximum measured imperfection value (amax) was therefore not

taken as amplitude of imperfection shape, but instead, the largest value in the form of the first

local buckling eigenmode (excluding the boundary measurements) was adopted as the most

suitable imperfection amplitude (au) as is shown in Fig. 5. The validity of this approach was

confirmed in a separate numerical study by the authors, featuring inelastic buckling of

isolated plates. A similar approach and similar findings were reported in [14].

The obtained imperfection amplitudes (Table 2) were compared with values generated from

Eq.(2), an expression originally developed by Dawson and Walker [17] and regarded as a

suitable general model for the prediction of initial imperfection amplitudes in simply

supported plates and in the plate elements of square hollow sections:

0.2/ /u cra t (2)

where au is the initial imperfection amplitude, t is the plate thickness, σ0.2 is the 0.2 % proof

strength of the material, σcr is the plate critical buckling stress and γ is a constant determined

for stainless steel box sections as γ = 0.023 [13, 14]. The experimental results are plotted

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against the predictions resulting from Eq. (2) in Fig. 6. The results obtained in the present

study suggest a higher value of γ = 0.045 which lies between the upper (γ = 0.111) and lower

(γ = 0.012) bounds proposed by Cruise and Gardner [16] following a detailed assessment of

imperfections in stainless steel members; this value is likely to be sensitive to the

manufacturing process and the length of specimen considered. Nonetheless, given the

inherent scatter associated with imperfections, the coefficient of determination R2 = 0.646

indicates the suitability of the Dawson and Walker model.

2.4 SHS stub column tests

Fourteen stub columns of six sizes of square hollow section (namely SHS 60×60×2, 80×80×2,

80×80×4, 100×100×3, 100×100×4 and 120×120×4) were tested. For two of the SHS

(100×100×3 and 120×120×4), one specimen was stress relieved (at 650°C) to remove residual

stresses. Measured geometric properties and ultimate loads for the stub columns are presented

in Table 3. The length of each stub column was equal to three times the nominal width of the

specimen, with the aim of eliminating the influence of global member buckling.

All tests were performed by means of a load-controlled hydraulic jack, with strain

stabilisation at several load levels in accordance with the recommendations of Galambos [18]

to achieve dynamic response free results. Loading rates were set such that ultimate load was

reached after approximately 50 to 60 minutes. End shortening of tested specimens was

recorded by three displacement transducers and four linear strain gauges located adjacent to

the corner regions of the sections. The test set-up can be seen in Fig. 7. The resulting load-end

shortening curves from the 14 stub column tests are presented in Fig. 8, with a typical failure

mode shown in Fig. 9.

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The structural response of the stress-relieved specimens exhibited some deviation from the as-

delivered specimens, but no definitive conclusions concerning the influence of residual

stresses on the structural response can be drawn since, even at the low annealing temperature

employed (650°C), a slight reduction in material yield strength also occurs. The influence of

residual stresses can however be isolated in finite element studies, as described in Section 3 of

this paper.

3. Numerical modelling

3.1 Development of FE models

In this section, the influence of residual stresses on structural response is studied numerically

by means of geometrically and materially non-linear FE models including imperfections

(GMNIA), using the FE package ABAQUS. For modelling of the thin-walled stainless steel

cross-sections the second order thin-shell element S9R5 with reduced integration and five

degrees of freedom per node was used. These elements are suitable for the modelling of thin

plates with thickness-to-width ratios less than 1/15, where transverse shear strains are small

[19]. This slenderness limit is fulfilled for all models developed in this study. Furthermore,

owing to the second order nature of the S9R5 element, hour-glassing, which is associated with

elements with reduced integration, does not occur, and the element is, in general,

computationally efficient. The adopted mesh, selected following a mesh sensitivity study, as

well as the boundary conditions of models can be seen in Fig.10. Measured material

properties were employed in the models, with enhanced material yield strength applied to an

extended corner area in accordance with [14].

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The residual stresses, together with the corresponding plastic strains, were included in the FE

models by means of the SIGINI and HARDINI Fortran subroutines [19]. In the SIGINI

subroutine, the values of the residual stresses are defined at the integration points; in this

study, six integration points were employed through the material thickness and the shape of

the residual stress distribution was defined by Gauss’ approximation, in place of the default

Simpson’s approximation. The advantage of an even number of integration points, which is

not permitted with Simpson’s approximation, is that the problem of a singularity at the mid-

thickness (with bending stresses reversing in sign) is not encountered, though a disadvantage

is that surface stresses are not recorded. The location of the integration points, together with

the applied residual stress distribution, is depicted in Fig. 11. For inelastic materials, in

addition to prescribing the residual stresses in the model, the accompanying plastic strains

(corresponding to the initial stress level 1) should also be defined by means of the HARDINI

subroutine. Since only uniaxial (longitudinal) residual stresses were considered, the

corresponding plastic strains could be calculated from the familiar Ramberg-Osgood

expression, given by Eq. (3).

1 1 1

0 0.2 0.2

0.002 0.002n n

pl elE

(3)

The membrane residual stresses were included in a similar way, following Eq. (4).

1 m b (4)

Owing to the non-linearity of the stress-strain relationships, the measured material response

from the coupon tests is affected by the initial deformation and bending residual stresses.

Indeed, bending residual stresses are inherently present in the stress-strain characteristics of

material extracted from structural sections (provided they are not straightened by plastic

deformation prior to testing). In this study, these effects were removed analytically, using an

iterative approach, such that residual stress free material characteristics could be ascertained.

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The analytical model comprised a series of layers through the thickness of the material and

considered plastic material response assuming von Mises yielding and Prandtl-Reuss

hardening, which is a simplification of the true material response of stainless steel behaviour

(neglecting, for example, anisotropy, non-symmetry and increases in ultimate strength due to

cold-working). Since the prescribed bending residual stresses in all layers were initially in

axial equilibrium, the mean stress (i.e. that corresponding to the residual stress containing

material) was simply calculated as an average stress from all the layers (5).

1

n

ii

n

(5)

Verification of the analytical model describing the stress-strain behaviour was achieved

through a simple FE model of a coupon test. A similar approach has been taken by other

authors [20].

3.2 Validation of FE models

The FE models were validated against the 14 stub column tests performed in the present

study, and two further tests – one stub column and one long column (2 m in length) – reported

by Gardner and Nethercot [21, 22]. For the 14 stub column tests in the present study, only

tensile material data were available, whilst for the two additional columns, compressive

material properties were available, together with specific residual stress measurements [7, 11].

Comparisons between test and FE results revealed that the ultimate load-carrying capacities of

the 14 stub columns from the present study were generally marginally under-predicted (by

approximately 6%) by the FE models – this is attributed to the use of tensile material

properties and the choice of magnitude of initial imperfections, which were based on

measured values (as described in Section 2.3), but applied in conjunction with the eigenmode

corresponding to the lowest elastic buckling load. Sensitivity analyses confirmed that ultimate

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load-carrying capacity was indeed relatively sensitive to the choice of imperfection amplitude.

In all cases, the general form of the load-end shortening response, as well as the failure mode,

were both well predicted by the numerical simulations. For the two additional columns [21,

22], where more precise input parameters were available, agreement between test and FE was

excellent, as shown in Fig.12 and 13. Fig. 12 shows the load-end shortening response of the

test and FE model of the SHS 1501504 stub column, while Fig. 13 shows the load-lateral

deflection response of a 2 m column of the same section. From the comparisons, it may be

concluded that the numerical models containing both components of residual stresses provide

the best fit to the experimental results, though sensitivity is not very pronounced.

3.3 Parametric studies

The following parametric studies examine the influence of residual stresses on the structural

response of stainless steel columns of varying global and local slenderness and are based on

previously described models of SHS 1501504 including the derived residual stress free

material properties. The magnitude and distribution of bending residual stresses was based on

the model presented in [11, 23] using 5% upper fractile values of stress magnitudes (Fig. 14).

It was proposed that these residual stresses have a rectangular bending stress block

distribution through the thickness of the section, which accords with the X-ray diffraction

measurements reported herein. The bending component of residual stress was used together

with measured membrane residual stresses from the SHS 150×150×4 (Fig. 15), since no

generalised model has been developed for membrane residual stresses in stainless steel box

sections.

3.3.1 Influence of residual stresses on global buckling

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By analysing column models of varying length, the influence of bending and membrane

residual stresses on global buckling capacity over a range of slendernesses was assessed. An

initial global imperfection amplitude of L/2000 (as adopted in previous investigations [14]),

where L is the column length, was employed throughout the present study. The results of the

study are presented in Fig. 16. For non-dimensional slendernesses (defined as the square

root of the ratio between yield load and elastic column buckling load) up to 1.4, the residual

stresses may be seen to have a positive influence on load-carrying capacity. Beyond this

slenderness, a negative influence is evident. Over the investigated range of slenderness,

inclusion of residual stresses causes a variation in resistance of between -2% and +10%. This

variation results principally from the effect of the bending residual stresses on the non-

linearity of the stress-strain curve. A positive influence of residual stresses arises when

column failure strains coincide with a region of increased tangent modulus. This is illustrated

in Fig. 17 where the material response with and without bending residual stresses is depicted.

Overall, the material stress-strain curve containing residual stresses may be seen to be

consistently below the residual stress free curve (i.e. the secant modulus is always lower).

However, the tangent modulus, which is known to be fundamental in controlling column

buckling resistance [24], is plotted in Fig. 18 for the stress free and residual stress containing

material. Below approximately 0.12% strain, the tangent modulus of the stress free curve is

higher than that of the residual stresses containing curve. Conversely, for higher strains, the

reverse is true. For non-dimensional column slenderness in the imperfection sensitive

region of between about 0.8 and 1.2, failure strains of approximately 0.15% were found; thus

the columns containing the bending residual stresses exhibited higher buckling loads. The

point where the tangent modulus curves of the stress free material and residual stress

containing material cross was found to be independent of the magnitude of residual stresses.

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For higher column slenderness ( > 1.5), lower strains are reached at ultimate load.

Therefore the tangent modulus for the material where bending residual stresses were included

is lower than the tangent modulus of material without residual stresses (Fig. 18), and thus

residual stresses were found to lead to a reduction in load-carrying capacity. However, the

structural behaviour of columns of high slenderness is less sensitive to the presence of both

geometric imperfections and residual stresses, so significant variation in capacity would not

be expected. The magnitude of residual stresses clearly influences the variation in load-

carrying capacity, and it was found that taking mean residual stresses values [11, 23] rather

than 5% upper fractile values, sensitivity of the column response was approximately halved.

3.3.2 Influence of residual stresses on local buckling

The influence of residual stresses on local buckling capacity was assessed in a similar manner

to above. Stub column models of varying local plate slenderness p (defined as the square

root of the yield load to the elastic local buckling load of the plate elements) with and without

residual stresses were examined. The results are shown in Fig. 19. The maximum influence of

residual stresses in terms of load-carrying capacity was 5.3%. Although less sensitive than the

column buckling results, similar conclusions can be drawn. The influence of membrane

residual stresses was found to be insignificant in comparison to the influence of the bending

component.

4. Conclusions

An experimental and numerical investigation of structural stainless steel hollow sections with

particular emphasis on the influence of residual stresses is described. X-ray diffraction was

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used for the detection of residual stress patterns in RHS elements. A thin layer of compressive

residual stresses was generally observed on the outer surface of the specimens – this is

believed to originate from the high contact forces from the forming tools. The remainder of

the outer half thickness of the section was in residual tension, and the through-thickness

measurements displayed a rectangular block-like residual stress pattern.

A total of 14 SHS stainless steel stub columns were tested under pure compression; the full

load-end shortening relationships were presented. The material properties of the flat and

corner areas of the sections were investigated and compared with existing predictive

formulae. Good agreement between the test results and predictions were observed. Initial

geometric plate imperfections were measured and compared with an existing predictive model

proposed by Dawson and Walker. A higher proportionality constant γ was found than that

presented by Gardner and Nethercot [14], but the value was within the range identified by

Cruise and Gardner [16]. Overall, the suitability of the Dawson and Walker formula was

clear.

The influence of bending and membrane residual stresses on global and local buckling was

investigated numerically through geometrically and materially non-linear FE analyses with

imperfections. The models were initially validated against experimental results.

Paradoxically, it was found that inclusion of residual stresses generally led to an increase in

load-carrying capacity. This was attributed principally to the influence of the bending residual

stresses on the material stress-strain curve. It was found that despite the secant modulus being

consistently reduced in the presence of residual stresses, the tangent modulus was increased in

some regions of the stress-strain curve. For cases where column failure strains coincided with

these increased tangent modulus regions (which was over the majority of the practical

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slenderness range), higher buckling loads resulted. Although the behaviour of stainless steel

columns with and without bending residual stresses has been investigated in this study, it

should be noted that these stresses will be inherently present in the stress-strain behaviour of

material extracted from structural sections, and would therefore not generally have to be

explicitly re-introduced into numerical models.

Acknowledgements

The authors would like to thank Dr Rachel Cruise from the University of Bath for valuable

consultations and for providing research results. The financial support of the Czech Ministry

of Education (Grant MSM 6840770001) is gratefully acknowledged.

References

[1] Euro-Inox and The Steel Construction Institute, Design manual for structural stainless

steel, Third ed. Brussels: Euro-Inox; 2006.

[2] Gardner L. The use of stainless steel in structures. Progress in Structural Engineering and

Materials 2005; 7(2), 45-55.

[3] Gardner L. Stainless steel structures in fire. Proceedings of the Institution of Civil

Engineers - Structures and Buildings. 2007; 160(3), 129-138.

[4] Machacek J. Study on imperfections of stiffened plating. Acta technica CSAV 1988;

33(5), 582-606.

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[5] Nethercot DA. Residual stresses and their influence upon the lateral buckling of rolled

steel beams. The Structural Engineer, 1974; 3(52), 89-96.

[6] Key PW, Hancock GJ. A theoretical investigation of the column behaviour of cold-formed

square hollow sections. Thin-Walled Structures 1993; 16(1-4), 31-64.

[7] Cruise RB, Gardner L. Residual stress analysis of structural stainless steel sections. J

Construct Steel Res 2008; 64(3), 352-366.

[8] Young B, Lui WM. Behaviour of cold-formed high strength stainless steel sections, J

Structural Engineering 2005; 131(11), 1738-45.

[9] Jandera M, Machacek J. Residual stresses and strength of hollow stainless steel sections.

In: Proc. 9th International Conference Modern Building Materials, Structures and Techniques,

Vilnius. 2007. 262-263 + CD

[10] Society for Experimental Mechanics. Handbook of Measurement of Residual Stresses.

The Fairmont Press Inc.; 1998.

[11] Cruise RB. The influence of production route on the response of structural stainless steel

members. Ph.D. thesis. Department of Civil and Environmental Engineering, Imperial

College London. 2007.

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[12] Gardner L, Ashraf M. Structural design for non-linear metallic materials. Engineering

Structures. 2006; 28(6), 926-934.

[13] Gardner L. A new approach to structural stainless steel design. Ph.D. thesis. Department

of Civil and Environmental Engineering, Imperial College London. 2002.

[14] Gardner L, Nethercot DA. Numerical modelling of stainless steel components – A

consistent approach. J Structural Engineering, ASCE, 2004; 130(10), 1586-1601.

[15] Ashraf M, Gardner L, Nethercot DA. Strength enhancement of the corner regions of

stainless steel cross-section. J Construct Steel Res, 2005; 61(1), 37-52.

[16] Cruise RB, Gardner L. Measurement and prediction of geometric imperfections in

structural stainless steel members. Structural Engineering and Mechanics. 2006; 24(1), 63-89.

[17] Dawson RG, Walker AC. Post-buckling of geometrically imperfect plates. J Struct Div.,

ASCE, 1972; 98(1), 75-94.

[18] Galambos TV. Guide to stability design criteria for metal structures. Fifth ed. New York:

John Wiley & Sons Inc.; 1998.

[19] ABAQUS. Analysis User's Manual, Volumes I.-VI., Version 6.5, Pawtucket, USA:

Hibbitt, Karlsson & Sorensen, Inc.; 2004.

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[20] Quach WM. Residual stresses in cold-formed steel sections and their effect on column

behaviour, PhD thesis, Hong Kong Polytechnic University, 2005.

[21] Gardner L, Nethercot DA. Experiments on stainless steel hollow sections – Part 1:

Material and cross-sectional behaviour. J Construct Steel Res, 2004; 60(9), 1291-1318.

[22] Gardner L, Nethercot DA. Experiments on stainless steel hollow sections – Part 2:

Member behaviour of columns and beams. J Construct Steel Res, 2004; 60(9), 1319-1332.

[23] Gardner L, Cruise RB. Modeling of residual stresses in structural stainless steel sections.

Journal of Structural Engineering, ASCE. In press.

[24] Allen HG, Bulson PS. Background to Buckling. McGraw-Hill Inc. London. 1980.

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Fig. 1: X-ray diffraction measurement apparatus

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Fig. 2: Outer surface residual stresses of RHS 100×80×2 in a) longitudinal and b) transverse

directions (tensile stresses are positive)

Fig. 3: Half through-thickness residual stress measurements in weld area of RHS 100×80×2

in a) longitudinal and b) transverse directions (tensile stresses are positive)

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Fig. 4: Prediction of 0.2% proof strength of corner material σ0.2,c from ultimate strength of

flat material σu,f

Fig. 5: Geometric imperfection measuring jig and evaluation

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Fig. 6: Analysis of initial geometric imperfection data

Fig. 7: Stub column test set-up

  23

Fig. 8: Stub column load-end shortening curves

Fig. 9: Typical stub column failure mode

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Fig. 10: Mesh and boundary conditions of presented FE models

Fig. 11: Initial through-thickness residual stress distribution employed in the FE models for

the parametric studies (bending residual stresses proposed by Cruise and Gardner [11,23])

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Fig. 12: Comparison of presented FE model with SHS 150×150×4 stub column test [21]

using membrane and bending residual stresses measured by Cruise and Gardner [7]

Fig. 13: Comparison of presented FE model with 2 m SHS 150×150×4 long column test [22]

using membrane and bending residual stresses measured by Cruise and Gardner [7]

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Fig. 14: 5% upper fractile values of bending residual stress magnitudes for cold-rolled box

sections proposed by Cruise and Gardner [7,23]

Fig. 15: Measured membrane residual stress distribution for SHS 1501504 [7]

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Fig. 16: Parametric study of residual stress influence on column load-carrying capacity

Fig. 17: Stress-strain relationship of material with and without residual stresses; result of

analytical model for material from the flat part of SHS 1501504

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Fig. 18: Analytically-generated tangent modulus of material with and without residual

stresses for SHS 1501504; the depicted through-thickness stress distributions relate to the

circled points for the bending residual stress containing material

Fig. 19: Parametric study of residual stress influence on stub column load-carrying capacity

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Table 1 Measured material properties for flat (F) and corner (C) areas of stainless steel

sections

Specimen E0 σ0.2 σ1.0 σu n n'0.2,1.0

[MPa] [MPa] [MPa] [MPa]

SHS 60×60×2-F 198000 430 470 789 7.2 2.8

SHS 60×60×2-C 194000 774 807 875 7.3 3.1

SHS 80×80×2-F 192000 427 480 785 4.9 3.0

SHS 80×80×2-C 201000 613 705 783 6.5 3.2

SHS 80×80×4-F 192000 435 498 789 4.2 3.0

SHS 80×80×4-C 221000 633 724 821 7.0 2.9

SHS 100×100×3-F 205750 417 457 753 7.1 2.3

SHS 100×100×3-C 202000 623 720 816 6.1 3.3

SHS 100×100×4-F 195000 430 477 736 4.1 2.6

SHS 100×100×4-C 200500 681 794 853 6.1 3.2

SHS 120×120×4-F 192000 429 479 783 4.3 2.7

SHS 120×120×4-C 212500 522 617 745 5.4 3.0

Table 2 Initial geometric imperfection measurement

Specimen Geometric imperfection values

amax [mm] au [mm]

SHS 60x60x2A 0.35 0.036

SHS 60x60x2B 0.42 0.039

SHS 80x80x2A 0.41 0.048

SHS 80x80x2B 0.58 0.136

SHS 80x80x4A 0.40 0.038

SHS 80x80x4B 0.44 0.034

SHS 100x100x3A 0.55 0.071

SHS 100x100x3B 0.41 0.061

SHS 100x100x3C* 0.62 0.075

SHS 100x100x4A 0.50 0.039

SHS 100x100x4B 0.50 0.044

SHS 120x120x4A 0.74 0.066

SHS 120x120x4B 0.77 0.103

SHS 120x120x4C* 0.88 0.100

* Stress relieved section

  30

Table 3 Measured geometry and key test results for SHS stub column test specimens 

Specimen Length

L Depth

D Breath

BThickness

t Outer cornerArea

AUltimate

load End

shortening

[mm] [mm] [mm] [mm]radius ro

[mm][mm2

] Fu [kN] at Fu [mm]

SHS 60x60x2A 180 60.06 60.14 2.22 2.21 528 274 2.32

SHS 60x60x2B 180 60.07 60.10 2.11 2.25 506 260 1.61

SHS 80x80x2A 240 79.86 79.92 1.86 2.55 598 206 0.79

SHS 80x80x2B 240 79.76 80.04 1.82 2.40 585 202 1.01

SHS 80x80x4A 240 80.28 80.41 3.88 6.03 1285 725 4.62

SHS 80x80x4B 240 80.17 80.42 3.80 5.98 1260 700 3.24SHS 100x100x3A 300 99.91 100.00 3.04 4.44 1234 550 1.73SHS 100x100x3B 300 99.91 100.10 3.00 4.47 1219 502 1.35SHS 100x100x3C* 300 99.98 100.11 3.08 4.30 1249 500 1.23SHS 100x100x4A 300 99.86 99.92 3.69 5.20 1498 776 2.87SHS 100x100x4B 300 99.86 99.92 3.69 5.20 1498 775 3.02SHS 120x120x4A 360 119.91 119.98 3.66 7.09 1823 775 2.01SHS 120x120x4B 360 119.93 120.02 3.68 7.06 1834 825 2.72SHS 120x120x4C* 360 119.96 120.01 3.67 7.09 1828 776 2.52

* Stress relieved sections