Structural performance of stainless steel circular hollow sections under

combined axial load and bending – Part 2: Parametric studies and design

Ou Zhao *a, Leroy Gardner b, Ben Young c

a, b Dept. of Civil and Environmental Engineering, Imperial College London, London, UK

c Dept. of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

* Corresponding author, Phone: +44 (0)20 7594 6058

Email: ou.zhao11@imperial.ac.uk

Keywords: Continuous strength method; Design standards; Finite element analysis;

Parametric studies; Reliability analysis; Stainless steel; Strain hardening; Structural design

Abstract

This paper reports the second part of the study on the structural behaviour of stainless steel

circular hollow sections subjected to combined axial load and bending moment. A series of

numerical parametric studies is presented, using the validated finite element (FE) models

from the companion paper, with the aim of generating further structural performance data

over a wider range of stainless steel grades, cross-section slendernesses and combinations of

loading. The considered parameters include the outer cross-section diameter, the ratio of

outer cross-section diameter to thickness and the initial loading eccentricity. Both the

experimentally and numerically derived section capacities were compared with the strength

predictions determined from the current European code, the American specification and the

Zhao, O., Gardner, L., & Young, B. (2016). Structural performance of stainless steel circular

hollow sections under combined axial load and bending – Part 2: Parametric studies and

design. Thin-Walled Structures, 101, 240-248.

Australian/New Zealand standard, allowing the applicability of each codified method to be

assessed. The comparisons revealed that the current design standards generally result in

unduly conservative and scattered strength predictions for stainless steel circular hollow

sections under combined loading, which can be primarily attributed to the neglect of strain

hardening in the determination of cross-section resistances and to the use of linear interaction

formulae. To overcome these shortcomings, improved design rules are proposed through

extension of the deformation-based continuous strength method (CSM) to the case of circular

hollow sections subjected to combined loading. Comparisons between the proposals and the

test and FE results indicate a high level of accuracy and consistency in the predictions. The

reliability of the proposed approach was confirmed by means of statistical analyses according

to EN 1990.

1. Introduction

Cold-formed stainless steel structural members are gaining increasing use in a range of

construction applications due to their aesthetic appeal, favourable mechanical properties and

excellent resistance against corrosion and fire. Given the high initial cost of stainless steels,

structural design efficiency is of primary concern. This has prompted research aimed at

assessing the accuracy of existing codes and developing new efficient design approaches for

stainless steel structures. With regards to cross-section load-carrying capacities, existing

design codes [1–3] generally limit the design stress to the 0.2% proof stress without

considering the pronounced strain hardening in the strength predictions of stocky cross-

sections, and neglect element interaction in the treatment of local buckling. A series of stub

column and four-point bending tests have been previously conducted on stainless steel closed

sections – square and rectangular hollow sections (SHS and RHS) [4–15] and circular hollow

sections (CHS) [16–22], and open sections – I-sections [16,18,23,24], channel sections

[18,25–28] and angle sections [18,29]. Comparisons of the test results with codified capacity

predictions revealed undue conservatism in the existing standards. To improve the design

efficiency, a deformation-based design approach called the continuous strength method

(CSM) [30–35], allowing a rational exploitation of strain hardening, has been proposed for

stocky cross-sections, and the Direct Strength Method (DSM) [36–38], accounting for the

beneficial effect of element interaction, was developed for slender cross-sections, both of

which significantly increase the material utilisation in structural design. Revised slenderness

limits for the classification of stainless steel cross-sections have also been proposed [39,40].

The structural behaviour of stainless steel SHS and RHS subjected to combined axial load

and bending moment has been systematically studied by Zhao et al. [41–43], where the

conservatism in existing codified design provisions was highlighted and improved design

rules were proposed, offering substantially enhanced capacity predictions.

The focus of the study in the present paper is on the structural performance of stainless steel

CHS under combined loading. Firstly, a series of parametric studies are reported, using the

finite element (FE) models validated in the companion paper [44], to expand the available

test data pool over a wider range of stainless steel grades, cross-section slendernesses and

combinations of loading. All the numerically derived data, together with the experimental

results, are then compared with the resistances predicted by EN 1993-1-4 [1], SEI/ASCE-8 [2]

and AS/NZS 4673 [3], enabling the accuracy of the existing codified methods to be evaluated.

Finally, improved design rules are sought through extension of the continuous strength

method to the case of stainless steel CHS under combined loading, and the applicability and

reliability of the method are carefully assessed.

2. Parametric studies

In this section, a series of parametric studies is presented, using the FE models validated in

the companion paper [44], aiming to extend the available structural performance data over a

wider range of stainless steel grades, cross-section slenderness and loading combinations. A

detailed description of the development of the FE models was given in the companion paper

[44], so only the key aspects relevant to the parametric studies are presented herein. The

parametric studies focus primarily on austenitic stainless steel, though comparative results are

also presented for duplex and ferritic grades. The adopted austenitic stainless steel stress–

strain curve was obtained from the tensile coupon tests on material cut from the CHS 76.3×3

specimens presented in the companion paper [44]. Since only austenitic material was tested,

the material properties for the parametric studies on the duplex and ferritic stainless steel

circular hollow sections were taken from previous tests on duplex and ferritic stainless steel

RHS under combined loading [41,43]. Table 1 reports the employed material properties for

each grade, where E is the Young’s modulus, σ0.2 is the 0.2% proof stress, σ1.0 is the 1.0%

proof stress, σu is the ultimate tensile strength, ε is a parameter defined as

0.2(235 / )( / 210000)E , and n, n’0.2,1.0 and n’0.2,u are the strain hardening exponents

used in the compound Ramberg–Osgood (R–O) material model [45–49]. Residual stresses

were not explicitly incorporated in the numerical models, as discussed in the companion

paper [44]. In terms of the geometric dimensions of the modelled circular hollow sections, the

outer diameter D was varied between 40 mm and 150 mm, while the cross-section thickness t

ranged from 0.7 mm to 10 mm. The resulting D/tε2 ratios varied between 15 and 88, covering

Class 1, 2 and 3 cross-sections, according to the slenderness limits in EN 1993-1-4 [1]. The

length of each model was set to be equal to three times the outer cross-section diameter. The

end section boundary conditions were applied by coupling all degrees of freedom of the end

section to an eccentric reference point, allowing only longitudinal translation and rotation

about the axis of buckling. The initial local geometric imperfection pattern along the member

length was assumed to be of the form of the lowest elastic buckling mode shape. The adopted

local geometric imperfection amplitude was taken as t/100, which was shown to lead to the

best agreement between the test and FE results in the model sensitivity study [44]. The initial

loading eccentricities ranged between 2 mm and 600 mm, leading to a wide range of loading

conditions being considered. In total, 472 results were generated [50], including 182 for

austenitic stainless steel, 145 for duplex stainless steel and 145 for ferritic stainless steel, all

of which are analysed and discussed in the following sections.

3 Assessment of codified design rules and development of new design methods

3.1 General

In this section, the codified design provisions for stainless steel circular hollow sections under

combined axial load and bending moment, as given in EN 1993-1-4 [1], SEI/ASCE-8 [2] and

AS/NZS 4673 [3], are firstly examined. Then, improved design rules are sought through

extension of the deformation-based continuous strength method (CSM) to the case of

combined loading, for which the development process is fully described. The accuracy of

each method is evaluated through comparisons of the ratios of test (or FE) to predicted

capacities under combined loading, calculated in terms of the axial load ratio, Nu/Nu,pred

[14,43], as reported in Table 2, where Nu is the test (or FE) axial load corresponding to the

distance on the N–M interaction curve from the origin to the test (or FE) data point (see Fig.

1), while Nu,pred is the predicted axial load corresponding to the distance from the origin to the

intersection with the design interaction curve, assuming proportional loading. A value of

Nu/Nu,pred greater than unity indicates that the test (or FE) data point lies outside the

interaction curve and is safely predicted. Note that all the comparisons are made based on the

unfactored design strengths.

3.2 European code EN 1993-1-4 (EC3)

The current European code for stainless steel, EN 1993-1-4 [1] adopts the same design

provisions for circular hollow sections under combined axial load and bending moment as

those given in EN 1993-1-1 [51] for carbon steel, where failure is determined based on a

linear summation of the utilization ratios under each component of loading, with a limit of

unity. The design expression is given by Eq. (1), in which NEd is the design ultimate axial

load, MEd is the design bending moment equal to the product of the design axial load NEd and

the sum of the initial loading eccentricity e0 and the generated mid-height lateral deflection at

failure e’, and NRd and MRd are the design values of the cross-section resistances under the

isolated loading conditions of pure compression and bending, respectively. NRd and MRd

depend on the cross-section classification: for Class 1 and 2 cross-sections, NRd is equal to the

yield load, defined as the product of the gross cross-section area A and the 0.2% proof stress

σ0.2, and MRd is given by the plastic moment capacity Mpl equal to the plastic section modulus

Wpl multiplied by σ0.2; for Class 3 cross-sections, NRd remains equal to the yield load, while

MRd reduces to the elastic moment capacity Mel, defined as the elastic section modulus Wel

multiplied by σ0.2; for Class 4 sections, the effective cross-section properties (Aeff and Weff) are

employed in place of the gross cross-section properties in the determination of NRd and MRd.

1Ed Ed

Rd Rd

N M

N M (1)

The combined loading test and FE results, normalised by the respective yield loads and

plastic moment capacities and arranged by cross-section class, are shown in Figs 2(a)–2(c)

for austenitic, duplex and ferritic stainless steels, respectively, together with the average

(since the Wel /Wpl ratio varied between sections) codified linear interaction curves. The

comparisons generally reveal a high level of scatter, and increasing conservatism of the EC3

strength predictions for all the material grades as the cross-sections become stockier (i.e.

moving from Class 3 to Class 1). A quantitative evaluation of the EN 1993-1-4 [1] capacity

predictions may be found in Table 2, which shows that the mean ratios of test (or FE) to EC3

failure loads Nu/Nu,EC3 are equal to 1.54, 1.43 and 1.31, with the coefficient of variations

(COV) equal to 0.14, 0.09 and 0.09, for the austenitic, duplex and ferritic stainless steel CHS

under combined axial load and bending moment, respectively. The rather conservative and

scattered nature of the EC3 predictions stems principally from the use of linear interaction

curves, which ignores the favourable spread of plasticity and stress redistribution within

stocky cross-sections, and the neglect of the pronounced strain hardening exhibited by

stainless steels, which limits the predictions of the end points of the design interaction curves

to the elastic or plastic moment capacities and yield load.

3.3 American specification SEI/ASCE-8

The American specification SEI/ASCE-8 [2] employs the same set of interaction formulae for

the design of both short (cross-section capacity) and long (global buckling) beam-columns, as

given by Eq. (2), in which Nn and Mn are the codified cross-sectional compression and

bending resistances, and Cm and αn are the equivalent moment factor and magnification factor,

respectively, both of which are approximately equal to unity for a short beam-column under

constant first order bending moment, as discussed by Zhao et al. [41]. Thus, Eq. (2) reduces

to the linear interaction formula given by Eq. (3), which is similar to the EC3 design

expression, except for the calculation of cross-section compression and bending resistances.

Note that the SEI/ASCE-8 [2] applies to cylindrical tubular sections with the outer diameter

to thickness ratio (D/t) less than 0.881E/σ0.2.

1Ed m Ed

n n n

N C M

N M (2)

1Ed Ed

n n

N M

N M (3)

The cross-section resistance in bending nM is calculated according to Eq. (4). Note that the

SEI/ASCE-8 provisions neglect plasticity in the determination of the cross-section bending

capacity and limit the maximum achieved cross-section bending capacity to the elastic

moment capacity for stocky sections with D/t ratios less than 0.112E/σ0.2.

el

n

c el

MM

K M

for 0.2

0.2 0.2

/ 0.112 /

0.112 / / 0.881 /

D t E

E D t E

(4)

in which cK is the reduction factor defined by Eq. (5),

0.21 / 5.882

18.93 / 8.93

c

c c

C E CK

D t

(5)

where C is the ratio of the effective proportional limit of the material to its yield (0.2% proof)

strength, and c is equal to 3.084C.

The cross-section compression resistance nN is determined from Eq. (6), where eA is the

effective cross-section area, as defined by Eq. (7), in which tE is the tangent modulus

corresponding to the 0.2% proof stress.

0.2n eN A (6)

2

1 1 1te c

EA K A

E

(7)

The accuracy of the American specification was evaluated by comparing the test and FE

results with the ASCE strength predictions. As can be seen from Table 2, the mean test (or

FE) to predicted failure load ratios Nu/Nu,ASCE are 1.78, 1.65 and 1.51, with corresponding

COVs of 0.19, 0.11 and 0.09 for the austenitic, duplex and ferritic stainless steels,

respectively, showing that the American specification SEI/ASCE-8 [2] results in greater

conservatism and higher scatter than the European code EN 1993-1-4 [1].

3.4 Australian/New Zealand standard AS/NZS 4673

The design rules for stainless steel circular hollow sections under combined loading in the

Australian/New Zealand standard AS/NZS 4673 [3] are the same as those in the American

specification SEI/ASCE-8 [2], except that AS/NZS 4673 [3] accounts for plasticity in the

calculation of cross-section bending capacities for stocky cross-sections and employs an

alternative reduction factor in the determination of bending capacities for slender cross-

sections. According to Clause 3.6.2 of AS/NZS 4673 [3], the full plastic moment capacity

may be used for cross-sections with D/t ratios less than 0.078E/σ0.2 (corresponding to a Class

2 slenderness limit in bending of 2/ 70D t , following the Eurocode format), the elastic

moment capacity may be used for sections with D/t ratios greater than 0.078E/σ0.2 but less

than 0.31E/σ0.2 (corresponding to a Class 3 slenderness limit in bending of 2/ 278D t ),

and a reduced elastic moment capacity may be used for slender cross-sections with D/t ratios

greater than 0.31E/σ0.2, as given by Eq. (8),

pl

a el

a el

M

M M

K M

for

0.2

0.2 0.2

0.2 0.2

/ 0.078 /

0.078 / / 0.31 /

0.31 / / 0.881 /

D t E

E D t E

E D t E

(8)

where Ma is the AS/NZS design cross-sectional bending resistance and Ka is a reduction

factor for local buckling, as determined from Eq. (9).

0.21 / 0.178

13.226 / 3.226

a

c c

C E CK

D t

(9)

It should be highlighted that the AS/NZS slenderness limits for Class 2 and 3 cross-sections

in bending are approximately equal to those set out in EN 1993-1-4 [1]. Given that a linear

interaction curve is also adopted in both codes, the Australian/New Zealand standard

AS/NZS 4673 and European code EN 1993-1-4 yield very similar capacity predictions for

non-slender stainless steel circular hollow sections under combined loading. The accuracy of

the AS/NZS 4673 [3] provisions is assessed in Table 2, showing that the mean ratios of test

(or FE) to AS/NZS predicted capacities Nu/Nu,AS/NZS are equal to 1.54, 1.43 and 1.31, with

COVs of 0.14, 0.09 and 0.09, for the austenitic, duplex and ferritic stainless steel CHS under

combined loading, respectively, which are the same values as derived from EN 1993-1-4 [1].

3.5 Continuous Strength Method (CSM)

The conservatism shown by the current codified design provisions for stainless steel circular

hollow sections under combined loading results principally from (1) the use of linear

interaction curves, which ignore the favourable effects of the spread of plasticity and stress

redistribution within cross-sections, and (2) the inaccurate predictions of the end points of the

interaction curves (i.e. the cross-section resistances under pure compression and bending),

which are determined without considering the influence of strain hardening.

The continuous strength method (CSM) is a deformation-based design approach [30–34],

which relates the strength of a cross-section to its deformation capacity and employs a bi-

linear material model to consider strain hardening. The application of the CSM has recently

been extended to circular hollow sections [35], showing a high level of accuracy and

consistency in the predictions of stainless steel cross-section resistances under compression

and bending, acting in isolation. Thus, improved design rules for CHS under combined

loading may be sought through the adoption of the CSM bending and compression

resistances as the end points and then employment of efficient nonlinear interaction curves. A

brief summary of the CSM for circular hollow sections subjected to isolated loading, and its

extension to the case of combined loading is described herein.

The application of the deformation-based CSM firstly requires identification of the

deformation capacity of the cross-section under the applied loading conditions. This may be

determined from the CSM ‘base curve’, which defines the relationship between the maximum

strain that a cross-section can endure and its local slenderness, as given by Eq. (10) [35],

3

4.5

4.44 10csm

y c

but 1min 15, u

y

C

(10)

where εcsm is the maximum attainable strain of the cross-section under the applied loading,

εy=σ0.2/E is the yield strain, and c is the local cross-section slenderness, calculated as

0.2 / cr , in which cr is the elastic local buckling stress of the cross-section, and is

calculated from Eq. (11) for a CHS under both compression or bending [52–54], and thus

also combined loading, in which υ is the Poisson’s ratio.

2

2

3 1cr

E t

D

(11)

The CSM elastic, linear hardening material model, which features four material parameters

(C1, C2, C3 and C4), is illustrated in Fig. 3, with the strain hardening slope Esh determined

from Eq. (12). The CSM material model parameter C1 is employed in Eq. (10) to prevent

over-predictions of strength from the linear hardening material model, with a value of 0.1 for

austenitic and duplex stainless steels and 0.4 for ferritic stainless steel. The CSM material

parameter C2 is used in Eq. (12) to define the strain hardening slope Esh, and is equal to 0.16

for austenitic and duplex stainless steels and 0.45 for ferritic stainless steel when εy/εu is less

than 0.45; when εy/εu is greater than or equal to 0.45, the strain hardening slope Esh is

assumed to be zero. The parameter εu=C3(1–σ0.2/σu)+C4 is the predicted strain corresponding

to the material ultimate strength, where C3 is equal to 1.0 for austenitic and duplex stainless

steels and 0.6 for ferritic stainless steel; C4 is equal to zero for all stainless steels.

0.2

2

ush

u y

EC

(12)

Eq. (10) applies for cross-section slenderness values less than or equal to 0.3, beyond which

the maximum attainable strain is less than the yield strain (i.e. εcsm/εy<1) and no significant

benefit arises from strain hardening. Beyond 0.3c , a base curve for slender cross-sections

is also provided in [35], but is not utilised herein since non-slender sections are the focus of

the present study.

Upon determination of the maximum attainable strain εcsm and the strain hardening modulus

Esh, the CSM design stress can then be calculated from Eq. (13), while the CSM resistances

for CHS subjected to pure compression (Ncsm) and pure bending (Mcsm) are determined from

Eqs (14) and (15) [32,55], respectively.

0.2csm sh csm yE (13)

csm csmN A (14)

2

1 1 1 /sh el csm el csmcsm pl

pl y pl y

E W W

EM M

W W

(15)

Figs 4(a)–4(c) depict the test and FE results normalised by the CSM compression and

bending resistances, indicating that the normalised test and FE data points now follow a

significantly tighter trend, in contrast to the rather scattered results when normalised by the

plastic moment capacity and yield load in Figs 2(a)–2(c). Thus, the adoption of the CSM

compression and bending resistances as the end points of the CHS interaction curves

substantially reduces the scatter and conservatism of the predictions.

Previous research [56] has shown that, assuming fully plastic behaviour, the theoretical

ultimate interaction relationship for CHS under combined axial compression and bending

moment may be defined by Eq. (16), in which MR,pl is the reduced plastic moment capacity

due to the existence of the applied axial force NEd and n is the ratio of the axial force to the

yield load NEd/Aσ0.2. It was also shown that Eq. (16) can be accurately approximated by the

simplified expression of Eq. (17). Fig. 5 shows the interaction curves determined from both

expressions. The comparison confirms that the simplified design expression closely follows

the theoretical solution. Considering the general distribution of the normalised test and FE

points in Figs 4(a)–4(c), it is therefore proposed to adopt the nonlinear form of the interaction

curve given by Eq. (17) for stainless steel, but with CSM compression and bending

resistances rather than the yield load and plastic moment capacity as the end points. Note that

a transition is defined at cross-section slenderness c equal to 0.27, corresponding to strain

ratio εcsm/εy of 1.6, beyond which a linear interaction curve is used in order to ensure

compatibility with the increasingly elastic end points as the cross-section slenderness

approaches 0.3c , where the CSM compression and bending resistances are equal to Aσ0.2

and Mel, respectively. The proposed CSM formulae for CHS under combined loading are thus

given by Eq. (18) for 0.27c and Eq. (19) for 0.27c , where MR,csm is the reduced CSM

bending moment resistance and ncsm is equal to the ratio of the design axial force to the CSM

compression resistance. The proposed interaction curves are compared against the test and FE

results in Figs 4(a)–4(c) for the three considered stainless steel material grades, where they

may be seen to accurately represent the general distribution of the test and FE data points.

,

0.2

cos2

R pl Ed

pl

M N

M A

(16)

1.7

, 1.04 1pl plR MM n , but plM (17)

,Rd csmE MM , where 1.7

, 0 11. 4 cR csm m ms csM nM , but csmM for 0.27c (18)

1Ed Ed

csm csm

N M

N M for 0.27c (19)

The mean ratios of test (or FE) to CSM predicted failure loads, Nu/Nu,csm, as reported in Table

2, are equal to 1.17, 1.15 and 1.10, with the corresponding COVs equal to 0.08, 0.07 and 0.07,

for the austenitic, duplex and ferritic stainless steel CHS subjected to combined axial load

and bending moment, respectively. Compared to the three codified design provisions,

considerable improvements in both the mean ratio of test (or FE) to predicted capacities and

the corresponding COV are achieved using the CSM. The improved accuracy and

consistency of the proposals may also be seen in Figs 6–8, where the CSM and codified

strength predictions are plotted against the test (or FE) results.

3.6 Reliability analysis

In this section, the reliability of the proposed CSM design approach for stainless steel circular

hollow sections under combined loading is demonstrated through statistical analyses,

conducted according to the provisions of EN 1990 [57]. Table 3 summarises the key

calculated statistical parameters for the CSM. The definitions of the parameters are as follows:

n is the number of tests and FE simulations, kd,n is the design (ultimate limit state) fractile

factor, b is the average ratio of test (or FE) to design model resistance based on a least

squares fit to all data, Vδ is the COV of the tests and FE simulations relative to the resistance

model, Vr is the combined COV incorporating both model and basic variable uncertainties,

and γM0 is the partial safety factor. In the analyses, the over-strength ratios for material yield

strength were taken as 1.3, 1.1 and 1.2, with COVs equal to 0.060, 0.030 and 0.045, for the

austenitic, duplex and ferritic stainless steels, respectively, while the COV of geometric

properties was taken as 0.050, as recommended by Afshan et al. [58]. The material over-

strength ratio is defined as the ratio of the mean value of yield strength produced by stainless

steel manufacturers to the value specified in EN 10088-4 [59]. Note that the values of the

adopted material over-strength ratios were derived by Afshan et al. [58], on the basis of

material data assembled from stainless steel producers.

It can be seen from Table 3 that the partial factors for the proposed CSM are equal to 0.92,

1.05 and 1.07 for austenitic, duplex and ferritic stainless steel CHS, respectively, which are

less than the currently adopted value of 1.1 in EN 1993-1-4, and thus suggest that the

presented CSM design proposals satisfy the reliability requirements of EN 1990 [57].

3.7 Summary

Overall, the three considered codified methods for the design of stainless steel circular hollow

sections under combined axial load and bending moment – the European code EN 1993-1-4

[1], the American specification SEI/ASCE-8 [2] and the Australian/New Zealand standard

AS/NZS 4673 [3], result in safe, but unduly conservative and scattered strength predictions,

primarily due to the neglect of strain hardening and the employment of linear interaction

design curves. The proposed CSM adopts a nonlinear interaction curve with CSM cross-

sectional compression and bending resistances as the end points. As shown in Table 2, the

CSM improves the mean ratio of test (or FE) to predicted capacities and the corresponding

COV by around 30% and 50%, respectively, compared to the codified design methods. Figs

9(a)–9(d) depict the combined loading test results compared against the four aforementioned

design interaction curves, also indicating the improved accuracy of the proposed approach.

4. Conclusions

Using the finite element models validated in the companion paper [44], parametric studies

were carried out to generate further structural performance data over a wider range of

stainless steel grades, cross-section slendernesses and combinations of compression and

flexural loading. The 472 generated FE results, together with the 23 experimental results

from the companion paper [44], were carefully analysed and then employed for the

assessment of the accuracy of the current European code EN 1993-1-4 [1], American

specification SEI/ASCE-8 [2] and Australian/New Zealand standard AS/NZS 4673 [3]. The

comparisons revealed that the codified capacity predictions for stainless steel circular hollow

sections subjected to combined compressive axial load and bending moment are unduly

conservative. This is mainly because of the employment of a linear interaction curve and the

adoption of conservative end points that ignore the beneficial influence of strain hardening.

The continuous strength method (CSM) is a deformation-based design approach that enables

a rational exploitation of strain hardening in the determination of cross-section resistances.

The proposed approach of using a nonlinear interaction curve, anchored to the CSM

compression and bending resistances, was shown to result in a high degree of accuracy and

consistency in the prediction of the resistances of stainless steel circular hollow sections

under combined loading. The reliability of the proposals was demonstrated by means of

statistical analyses according to the provisions of EN 1990 [57]. It is therefore recommended

that the proposed approach for circular hollow sections be considered for incorporation into

future revisions of stainless steel structural design standards.

Acknowledgements

The authors gratefully acknowledge the Joint PhD Scholarship from Imperial College

London and the University of Hong Kong for the financial support.

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Fig. 1. Definition of Nu and Nu,pred on moment–axial load interaction curve.

(a) Austenitic stainless steel

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

pl

Nu/Aσ0.2

EC3: Class 1, 2

EC3: Class 3

Class 1 sections

Class 2 sections

Class 3 sections

Design interaction

curve

Test (or FE) capacity

Predicted capacity

M

N Nu Nu,pred

(b) Duplex stainless steel

(c) Ferritic stainless steel

Fig. 2. Combined loading test and FE results normalised by the plastic moment capacity and yield load (i.e. the

EC3 end points for Class 1 and 2 cross-sections).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

pl

Nu/Aσ0.2

EC3: Class 1, 2

EC3: Class 3

Class 1 sections

Class 2 sections

Class 3 sections

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

pl

Nu/Aσ0.2

EC3: Class 1, 2

EC3: Class 3

Class 1 sections

Class 2 sections

Class 3 sections

Fig. 3. CSM elastic, linear hardening material model.

(a) Austenitic stainless steel

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

csm

Nu/Ncsm

Class 1 sections

Class 2 sections

Class 3 sections

CSM nonlinear design curve (Eq. (18))

CSM linear design curve (Eq. (19))

ε

Esh fy

εy

σ

fu

C1εu C2εu

E

3 41y

u

u

fC C

f

(b) Duplex stainless steel

(c) Ferritic stainless steel

Fig. 4. Combined loading test and FE results normalised by the CSM compression and bending resistances.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

csm

Nu/Ncsm

Class 1 sections

Class 2 sections

Class 3 sections

CSM nonlinear design curve (Eq. (18))

CSM linear design curve (Eq. (19))

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mu/M

csm

Nu/Ncsm

Class 1 sections

Class 2 sections

Class 3 sections

CSM nonlinear design curve (Eq. (18))

CSM linear design curve (Eq.(19))

Fig. 5. Comparison of simplified interaction curve with the theoretical design curve.

Fig. 6. Comparison of test and FE results with CSM and EN 1993-1-4 strength predictions.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

MR

,pl/M

pl

NEd/Aσ0.2

Theoretical Eq. (16)

Simplified Eq. (17)

0

200

400

600

800

1000

0 200 400 600 800 1000

Nu

,test o

r N

u,F

E (

kN

)

Nu,pred (kN)

CSM

EN 1993-1-4

Fig. 7. Comparison of test and FE results with CSM and SEI/ASCE-8 strength predictions.

Fig. 8. Comparison of test and FE results with CSM and AS/NZS strength predictions.

0

200

400

600

800

1000

0 200 400 600 800 1000

Nu

,test o

r N

u,F

E (

kN

)

Nu,pred (kN)

CSM

SEI/ASCE-8

0

200

400

600

800

1000

0 200 400 600 800 1000

Nu

,test o

r N

u,F

E (

kN

)

Nu,pred (kN)

CSM

AS/NZS 4673

(a) CHS 60.5×2.8 specimens.

(b) CHS 76.3×3 specimens.

(c) CHS 114.3×3 specimens.

(d) CHS 139.4×3 specimens.

Fig. 9. Comparison of combined loading test results with the four design interaction curves.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Mu/M

pl

Nu/Aσ0.2

Tests

EN 1993-1-4

SEI/ASCE-8

AS/NZS

CSM

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Mu/M

pl

Nu/Aσ0.2

Tests

EN 1993-1-4

SEI/ASCE-8

AS/NZS

CSM

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Mu/M

pl

Nu/Aσ0.2

Tests

EN 1993-1-4

SEI/ASCE-8

AS/NZS

CSM

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Mu/M

pl

Nu/Aσ0.2

Tests

EN 1993-1-4

SEI/ASCE-8

AS/NZS

CSM

EN1993-1-4 and AS/NZS

design curves coincide

EN1993-1-4 and AS/NZS

design curves coincide

EN1993-1-4 and AS/NZS

design curves coincide

Three codified design

curves coincide

Table 1 Summary of key measured material properties for the tensile coupons.

Grade E σ0.2 σ1.0 σu ε R-O coefficient

(GPa) (MPa) (MPa) (MPa)

n 0.2,1.0'n

0.2,' un

Austenitic 195 302 347 784 0.85 7.3 2.0 1.9

Duplex 199 519 578 728 0.65 5.3 2.8 3.7

Ferritic 190 466 508 515 0.68 6.6 7.6 7.6

Table 2 Comparisons of combined loading test and FE results with predicted strengths.

(a) Austenitic stainless steel

No. of tests: 23 Nu/Nu,EC3 Nu/Nu,ASCE Nu/Nu,AS/NZS Nu/Nu,csm

No. of FE simulations: 182

Mean 1.54 1.78 1.54 1.17

COV 0.14 0.19 0.14 0.08

(b) Duplex stainless steel

No. of tests: 0 Nu/Nu,EC3 Nu/Nu,ASCE Nu/Nu,AS/NZS Nu/Nu,csm

No. of FE simulations: 145

Mean 1.43 1.65 1.43 1.15

COV 0.09 0.11 0.09 0.07

(c) Ferritic stainless steel

No. of tests: 0 Nu/Nu,EC3 Nu/Nu,ASCE Nu/Nu,AS/NZS Nu/Nu,csm

No. of FE simulations: 145

Mean 1.31 1.51 1.31 1.10

COV 0.09 0.09 0.09 0.07

Table 3 Reliability analysis results calculated according to EN 1990.

Grade No. of tests and FE simulations kd,n b Vδ Vr γM0

Austenitic 205 3.138 1.188 0.081 0.112 0.92

Duplex 145 3.159 1.143 0.066 0.088 1.05

Ferritic 145 3.159 1.063 0.072 0.099 1.07