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: [email protected]
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.
References
[1] EN 1993-1-4. Eurocode 3: Design of steel structures – Part 1.4: General rules –
Supplementary rules for stainless steels. Brussels: European Committee for Standardization
(CEN); 2006.
[2] SEI/ASCE 8-02. Specification for the design of cold-formed stainless steel structural
members. Reston: American Society of Civil Engineers (ASCE); 2002.
[3] AS/NZS 4673. Cold-formed stainless steel structures. Sydney: AS/NZS 4673:2001; 2001.
[4] Rasmussen KJR, Hancock GJ. Design of cold-formed stainless steel tubular members. I:
Columns. Journal of Structural Engineering (ASCE), 1993;119(8):2349–67.
[5] Rasmussen KJR, Hancock GJ. Design of cold-formed stainless steel tubular members. II:
beams. Journal of Structural Engineering (ASCE), 1993;119(8):2368–86.
[6] Gardner L, Nethercot DA. Experiments on stainless steel hollow sections – Part 1:
Material and cross-sectional behaviour. Journal of Constructional Steel Research,
2004;60(9):1291–318.
[7] Gardner L, Nethercot DA. Experiments on stainless steel hollow sections – Part 2:
Member behaviour of columns and beams. Journal of Constructional Steel Research,
2004;60(9):1319–32.
[8] Zhou F, Young B. Tests of cold-formed stainless steel tubular flexural members. Thin-
Walled Structures, 2005;43(9):1325–37.
[9] Theofanous M, Gardner L. Testing and numerical modelling of lean duplex stainless steel
hollow section columns. Engineering Structures, 2009;31(12):3047–58.
[10] Theofanous M, Gardner L. Experimental and numerical studies of lean duplex stainless
steel beams. Journal of Constructional Steel Research, 2010;66(6):816–25.
[11] Huang Y, Young B. Experimental and numerical investigation of cold-formed lean
duplex stainless steel flexural members. Thin-Walled Structures, 2013;73(19):216–28.
[12] Afshan S, Gardner L. Experimental study of cold-formed ferritic stainless steel hollow
sections. Journal of Structural Engineering (ASCE), 2013;139(5):717–28.
[13] Shu G, Zheng B, Shen X. Experimental and theoretical study on the behaviour of cold-
formed stainless steel stub columns. International Journal of Steel Structures, 2013;13(1)141–
53.
[14] Zhao O, Gardner L, Young B. Buckling of ferritic stainless steel members under
combined axial compression and bending. Journal of Constructional Steel Research,
2016;117:35–48.
[15] Theofanous M, Saliba N, Zhao O, Gardner, L. Ultimate response of stainless steel
continuous beams. Thin-Walled Structures, 2014;83:115–27.
[16] Talja, A. Test report on welded I and CHS beams, columns and beam-columns. Report
to ECSC. VTT Building Technology, Finland; 1997.
[17] Young B, Hartono W. Compression tests of stainless steel tubular members. Journal of
Structural Engineering (ASCE), 2002;128(6):754–61.
[18] Kuwamura H. Local buckling of thin-walled stainless steel members. Steel Structures,
2003;3:191–201.
[19] Kiymaz G. Strength and stability criteria for thin-walled stainless steel circular hollow
section members under bending. Thin-Walled Structures, 2005;43(10):1534–49.
[20] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression – part
I: Experiments. International Journal of Mechanical Sciences, 2006;48(8):830–41.
[21] Lam D, Gardner L. Structural design of stainless steel concrete filled columns. Journal
of Constructional Steel Research, 2008;64(11):1275–82.
[22] Uy B, Tao Z, Han LH. Behaviour of short and slender concrete-filled stainless steel
tubular columns. Journal of Constructional Steel Research, 2011;67(3):360–78.
[23] Yuan HX, Wang YQ, Shi YJ, Gardner L. Stub column tests on stainless steel built-up
sections. Thin-Walled Structures, 2014;83:103–14.
[24] Saliba N, Gardner L. Cross-section stability of lean duplex stainless steel welded I-
sections. Journal of Constructional Steel Research, 2013;80:1–14.
[25] Lecce M, Rasmussen KJR. Distortional Buckling of Cold-Formed Stainless Steel
Sections: Experimental Investigation. Journal of Structural Engineering (ASCE), 2006;132(4):
497–504.
[26] Becque J, Rasmussen KJR. Experimental investigation of local-overall interaction
buckling of stainless steel lipped channel columns. Journal of Constructional Steel Research,
2009;65(8–9):1677–84.
[27] Rossi B, Jaspart JP, Rasmussen KJR. Combined Distortional and Overall Flexural-
Torsional Buckling of Cold-Formed Stainless Steel Sections: Experimental Investigations.
Journal of Structural Engineering (ASCE), 2010;136(4):354–60.
[28] Niu S, Rasmussen KJR, Fan F. Distortional–global interaction buckling of stainless steel
C-beams: Part I - Experimental investigation. Journal of Constructional Steel Research,
2014;96:127–39.
[29] Theofanous M, Liew A, Gardner L. Experimental study of stainless steel angles and
channels in bending. Structures, 2015;4:80-90.
[30] Ashraf M, Gardner L, Nethercot DA. Structural stainless steel design: resistance based
on deformation capacity. Journal of Structural Engineering (ASCE), 2008;134(3):402–11.
[31] Gardner L. The Continuous Strength Method. Proceedings of the Institution of Civil
Engineers – Structures and Buildings, 2008;161(3):127–33.
[32] Afshan S, Gardner L. The continuous strength method for structural stainless steel
design. Thin-Walled Structures, 2013;68(4):42–49.
[33] Liew A, Gardner L. Ultimate capacity of structural steel cross-sections under
compression, bending and combined loading. Structures, 2015;1:2–11.
[34] Ashraf M, Gardner L, Nethercot DA. Compression strength of stainless steel cross-
sections. Journal of Constructional Steel Research, 2006;62(1):105–15.
[35] Buchanan C, Gardner L, Liew A. The continuous strength method for the design of
circular hollow sections. Journal of Constructional Steel Research, 2015;118:206-217.
[36] Schafer BW. Review: the direct strength method of cold-formed steel member design.
Journal of Constructional Steel Research, 2008;64(7):766–78.
[37] Becque J, Lecce M, Rasmussen KJR. The direct strength method for stainless steel
compression members. Journal of Constructional Steel Research, 2008;64(11):1231–8.
[38] Rossi B, Rasmussen KJR. Carrying capacity of stainless steel columns in the low
slenderness range. Journal of Structural Engineering (ASCE), 2012;139(6):1088–92.
[39] Gardner L, Theofanous M. Discrete and continuous treatment of local buckling in
stainless steel elements. Journal of Constructional Steel Research, 2008;64(11):1207–16.
[40] Gardner L, Law KH, Buchanan C. Unified slenderness limits for structural steel circular
hollow sections. Romanian Journal of Technical Sciences, Applied Mechanics, 2014;59(1–
2):153–63.
[41] Zhao O, Rossi B, Gardner L, Young, B. Behaviour of structural stainless steel cross-
sections under combined loading – Part I: Experimental study. Engineering Structures,
2015;89:236–46.
[42] Zhao O, Rossi B, Gardner L, Young B. Behaviour of structural stainless steel cross-
sections under combined loading – Part II: Numerical modelling and design approach.
Engineering Structures, 2015;89:247–59.
[43] Zhao O, Rossi B, Gardner L, Young B. Experimental and numerical studies of ferritic
stainless steel tubular cross-sections under combined compression and bending. Journal of
Structural Engineering (ASCE), 10.1061/(ASCE)ST.1943-541X.0001366,04015110.
[44] Zhao O, Gardner L, Young B. Structural performance of stainless steel circular hollow
sections under combined axial load and bending – Part 1: Experiments and numerical
modelling. Thin-Walled Structures, submitted.
[45] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters.
Technical note No 902, Washington DC: National advisory committee for aeronautics; 1943.
[46] Hill HN. Determination of stress–strain relations from offset yield strength values.
Technical note No 927, Washington DC: National advisory committee for aeronautics; 1944.
[47] Mirambell E, Real E. On the calculation of deflections in structural stainless steel beams:
An experimental and numerical investigation. Journal of Constructional Steel Research,
2000;54(1):109–33.
[48] Rasmussen KJR. Full-range stress-strain curves for stainless steel alloys. Journal of
Constructional Steel Research, 2003;59(1):47–61.
[49] Gardner L, Ashraf M. Structural design for non-linear metallic materials. Engineering
Structures, 2006;28(6):926–34.
[50] Zhao O. Structural behaviour of stainless steel elements subjected to combined loading.
PhD Thesis, Department of Civil and Environmental Engineering, Imperial College London,
UK; 2015.
[51] EN 1993-1-1. Eurocode 3: Design of steel structures – Part 1.1: General rules and rules
for buildings. Brussels: European Committee for Standardization (CEN); 2005.
[52] Seide P, Weingarten VI. On the buckling of circular cylindrical shells under pure
bending. Journal of Applied Mechanics, 1961;28(1):112–16.
[53] Reddy BD, Calladine CR. Classical bucking of a thin-walled tube subjected to bending
moment and internal pressure. International Journal of Mechanical Sciences, 1978;20:641–50.
[54] Silvestre N. Generalised beam theory to analyse the buckling behaviour of circular
cylindrical shells and tubes. Thin-Walled Structures, 2007;45(2):185–98.
[55] Gardner L, Wang F, Liew A. Influence of strain hardening on the behavior and design of
steel structures. International Journal of Structural Stability and Dynamics, 2011;11(05):855–
75.
[56] Rondal J, Würker K-G, Dutta D, Wardenier J, Yeomans, N. Structural stability of hollow
sections, 1st Ed. Verlag TÜV Rheinland, Germany; 1992.
[57] EN 1990. Eurocode – basis of structural design. Brussels: European Committee for
Standardization (CEN); 2002.
[58] Afshan S, Francis P, Baddoo NR, Gardner L. Reliability analysis of structural stainless
steel design provisions. Journal of Constructional Steel Research, 2015;114:293–304.
[59] EN 10088-4. Stainless steels – Technical delivery conditions for sheet/plate and strip of
corrosion resisting steels for construction purposes. Brussels: European Committee for
Standardization (CEN); 2009.
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