the continuous strength method for the design of hot-rolled steel cross-sections · 2018. 12....
TRANSCRIPT
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Yun, X., Gardner, L. and Boissonnade, N. (2018) The Continuous Strength Method for The
Design of Hot-Rolled Steel Cross-Sections. Engineering Structures. 157, 179-191.
The continuous strength method for the design of
hot-rolled steel cross-sections
Xiang Yun a, Leroy Gardner b, Nicolas Boissonnade b
a Department of Civil and Environmental Engineering, Imperial College London, SW7 2AZ, UK.
Email: [email protected]
b Department of Civil and Environmental Engineering, Imperial College London, SW7 2AZ, UK.
Email: [email protected] (Corresponding author)
c Department of Civil and Water Engineering, Laval University, Quebec, Canada
Email: [email protected]
Abstract
The continuous strength method (CSM) is a deformation-based structural design method that
enables material strain hardening properties to be exploited, thus resulting in more accurate
and consistent capacity predictions. To date, the CSM has featured an elastic, linear hardening
material model and has been applied to cold-formed steel, stainless steel and aluminium.
However, owing to the existence of a yield plateau in its stress-strain response, this model is
not well suited to hot-rolled carbon steel. Thus, a tri-linear material model, which can closely
represent the stress-strain response of hot-rolled carbon steel, is introduced and incorporated
into the CSM design framework. Maintaining the basic design philosophy of the existing CSM,
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new cross-section resistance expressions are derived for a range of hot-rolled steel structural
section types subjected to compression and bending. The design provisions of EN 1993-1-1
and the proposed CSM are compared with experimental results collected from the literature
and numerical simulations performed in this paper. Overall, the CSM is found to offer more
accurate and consistent predictions than the current design provisions of EN 1993-1-1. Finally,
statistical analyses are carried out to assess the reliability level of the two different design
methods according to EN 1990 (2002).
Keywords: Continuous strength method, Finite element modelling, Hot-rolled carbon steel,
Reliability analysis, Strain hardening, Yield plateau
1. Introduction
The resistances of hot-rolled structural steel cross-sections are traditionally determined
following the process of cross-section classification, which is based on the assumption of
elastic, perfectly plastic material behaviour. Cross-section classification is a key feature of
many modern steel design codes, such as EN 1993-1-1 [1], and determines the extent to which
the resistance and deformation capacities of a cross-section are limited by the effects of local
buckling. The classification of a cross-section is assessed by comparing the width-to-thickness
ratios of its constituent plate elements to corresponding slenderness limits, which take account
of the edge support conditions and applied loading. The plate elements are typically treated
individually, thus neglecting the interaction effects between adjacent elements, i.e. the ability
of the less slender elements to provide some assistance in resisting local buckling to the more
slender elements, and the classification of the most slender element defines that of the overall
cross-section. Furthermore, the maximum stress in the cross-section is generally limited to the
material yield stress fy, neglecting the beneficial effects of strain hardening. Hence, for non-
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slender cross-sections the compression resistance is limited to the yield load Ny and the cross-
section bending resistance to the plastic moment capacity Mpl for Class 1 or Class 2 cross-
sections and the elastic moment capacity Mel for Class 3 cross-sections, which results in a step
(or recently proposed linear or parabolic transition [2,3]) from Mpl to Mel at a particular
slenderness limit. Experimental results have shown that the current design rules in EN 1993-1-
1 [1] are often conservative in estimating the resistance of stocky hot-rolled steel cross-sections
in both compression and bending due primarily to the omission of strain hardening [4-6], and
the failure to account accurately for the spread of plasticity in Class 3 cross-sections [7-8].
The importance of plate element interaction effects on the ultimate resistance of structural
cross-sections has been examined by a number of researchers. Studies have been conducted on
square and rectangular hollow sections (SHS/RHS) [9-12] and I-sections [12-16], while
explicit allowance for element interaction through the use of cross-section rather than element
elastic buckling stresses in the determination of local and distortional slendernesses is a key
feature of the Direct Strength Method (DSM) [17-19]. The beneficial influence of strain
hardening has also been observed by a number of researchers [6,20-22], and the exploitation
of strain hardening is a key feature of the Continuous Strength Method [23,24].
The CSM is a deformation-based design approach that offers more consistent and continuous
resistance predictions than traditional approaches based on cross-section classification, and
allows systematically for the beneficial influence of strain hardening. To date, this method has
been established for stainless steel [25-27], structural carbon steel [23,28-30] and aluminium
alloy design [31,32] utilizing a simple elastic, linear hardening material model, with a strain
hardening modulus Esh varying with material grade. This model has been shown to offer a
suitably close representation of the stress-strain responses of metallic materials with nonlinear
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rounded stress-strain behaviour, such as stainless steel, cold-formed carbon steel and
aluminium alloys, to enable efficient and accurate designs. However, due to the existence of a
yield plateau, this CSM bi-linear material model is less suitable for hot-rolled carbon steels.
Thus, a revised CSM material model has been proposed for hot-rolled carbon steels [33] that
exhibits a yield point, followed by a yield plateau and a strain hardening region. In this paper,
the application of the CSM to hot-rolled steel structural elements, i.e. SHS/RHS and I-sections,
focusing primarily on cross-sections in compression and bending, including the development
of the material model and the corresponding resistance functions, is outlined. The accuracy of
the current design methods in EN 1993-1-1 and the CSM are then assessed based on the results
of experiments collected from the literature and numerical simulations performed herein.
2. Numerical modelling
Numerical analyses were carried out using the finite element programme ABAQUS, version
6.13 [34] to simulate the cross-sectional response of hot-rolled steel SHS/RHS and I-sections
in both compression and bending. The numerical models were initially validated against
existing experimental results on stub columns [4,35-37] and simply supported beams [6,38],
and were subsequently used for parametric studies to expand the numerical data over a wider
range of cross-section slendernesses, section geometries and loading conditions. Details of the
finite element (FE) modelling approach and the parametric studies performed in the current
study are presented in the following subsections.
2.1. Description of the FE model
The reduced integration four-nodded doubly curved shell element S4R was employed in all FE
models; this element type has been successfully utilised by others in similar applications
[9,38,39]. Mesh sensitivity studies were conducted to determine an appropriate mesh density
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to provide accurate results while minimizing computational time. Hence, an average element
size equal to one twentieth of the largest plate width that makes up the cross-section was
adopted for the flat parts of the modelled cross-sections, while four elements were employed
to model the curved geometry of the corner regions for the SHS/RHS and two elements for
each fillet zone of the I-sections. For the I-sections, particular attention was given to ensure
that the properties of the fillet zones could be accurately represented. The nodes at each end of
the web were shifted by a distance of half the flange thickness to avoid overlapping of the
elements at the web-to-flange junction, and these nodes were tied to their corresponding nodes
at the mid-thickness of the flanges using the “General multi-point constraints (*MPC)”; this
ensured that the translational and rotational degrees of freedom were equal for this pair of
nodes. The additional area in the fillet zones was allowed for by increasing the thickness of the
adjacent web elements (see Fig. 1).
During the validation of the models, measured cross-section dimensions and material
properties from the existing tests were incorporated into the FE models to replicate the
observed experimental behaviour. In cases where the full stress-strain curves were not reported
in the literature, the nonlinear material model for hot-rolled carbon steels proposed in [33] was
used. Since the analyses may involve large inelastic strains, the engineering (nominal) stress-
strain curves were converted to true stress-logarithmic plastic strain curves as required in
ABAQUS for the chosen element type.
Initial geometric imperfections were incorporated into the FE models using the mode shapes
obtained from a prior elastic buckling analysis. The lowest buckling mode under the considered
loading conditions with an odd number buckling half-waves was used as the imperfection
shape, which generally represents the most unfavourable imperfection pattern [40], as shown
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in Fig. 2. Four different imperfection amplitudes were examined: a/400, a/200 and a/100,
where a is the flat width of the most slender constituent plate element in the cross-section (i.e.
that with the highest value of cry /f under the applied loading conditions), and a value
obtained from the modified Dawson and Walker predictive model [4,41], as given by Eq. (1),
where ω0 is the magnitude of the local imperfection, t is the plate thickness, fy is the material
yield stress, and σcr is the elastic buckling stress of the most slender cross-section plate element
assuming simply supported conditions between adjacent plates, taking due account of the stress
distribution through the buckling factor kσ [42].
tσ
fω
cr
y
0 064.0 (1)
The residual stresses introduced into hot-rolled steel sections are primarily associated with non-
uniform cooling rates during the fabrication process, with the faster cooling regions of the
sections being left in residual compression and the slower cooling regions in residual tension.
The residual stress patterns recommended by the European Convention for Constructional
Steelwork (ECCS) [43], as shown in Fig. 3, where compressive residual stresses are designated
as positive and tensile residual stresses as negative, were applied to the FE models for the I-
sections. The magnitude of the initial stress depends on whether the height-to-width ratio of
the cross-section is less than or equal to 1.2 or greater than 1.2 and is independent of the yield
stress, with the nominal stress fy* = 235 MPa taken as the reference value. For SHS and RHS,
a typical pattern of residual stresses, based on DIN recommendations [44], is shown in Fig.
4(a). To simplify the distribution for ease of application in FE modelling, a modified residual
stress pattern was proposed by Nseir [35] based on the analysis of measured residual stress
magnitudes and distributions, as represented in Fig. 4(b). A constant value of 0.5fy* was
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assumed for the compressive residual stresses in the corner regions while the amplitudes of the
tensile residual stresses in the flat regions of the web and flanges were determined to achieve
self-equilibrium. The simplified residual stress pattern was employed herein for both the
validation of the models and the parametric analyses.
The boundary conditions were carefully selected to simulate the experimental setups. For the
stub column FE models, each end section was connected to a concentric reference point through
rigid body constraints such that the degrees of freedom of all nodes at each end were
constrained to the degrees of freedom of its corresponding reference point. All degrees of
freedom were then restrained at each end apart from the longitudinal translation at the loaded
end to allow for vertical displacement. For the beam models, the reference points were
positioned on the lower flange of the section and connected to the nodes within the
corresponding end plate region, with boundary conditions applied at each reference point to
allow appropriate movement and rotation to simulate simple support conditions. The point
loads were applied to the lower part of the web at the web-to-flange junction to prevent web
crippling under concentrated loading. Also, for modelling convenience, beam models under
pure bending moment were developed and validated against 4-point bending tests and then
used in the subsequent parametric studies. The length of the pure bending models was equal to
the length of the constant moment region in the corresponding 4-point bending tests, and
similar boundary conditions as those used for the stub column models were applied to the pure
bending models, the only difference being that the reference points were allowed to rotate in
the direction of bending. The modified Riks method was employed to perform the
geometrically and materially nonlinear analyses of the FE models to capture the full load-
deformation response of the modelled specimens, including the post-ultimate path.
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2.2. Validation of the FE models
The FE models were validated by comparing the numerically obtained ultimate capacities,
load-deformation responses and the deformed shapes against those reported in a series of
experiments. The normalized results obtained from the FE models for varying imperfection
magnitudes for stub columns [4,35-37] and beams [6,38] are presented in Tables 1 and 2,
respectively. It was generally observed that, for both compression and bending resistances, the
most consistent predictions were obtained using the local imperfection amplitude of a/200,
which is also in accordance with the recommendations of EN 1993-1-5 [42]. Note however
that the results of the FE simulations were relatively insensitive to the magnitude of the
imperfections, especially for the more stocky cross-sections. The pure bending models were
found to provide accurate and consistent predictions compared with the corresponding tests
and 4-point bending models and were thus considered to be appropriate to perform parametric
studies for bending. The load-deformation curves obtained from the FE models with the local
imperfection amplitude of a/200 are compared against the corresponding tests results for
typical stub columns and beams in Figs 5-8, where δ is the end shortening of the stub columns,
κ is the curvature of the beams, κpl is the elastic portion of the total curvature corresponding to
the plastic moment Mpl and θ is the end rotation of the beams. The initial stiffnesses, ultimate
capacities and general shape of the load-deformation histories from the FE models also
matched closely with those obtained from the tests, and the failure modes of the FE models
mirrored those observed in the physical experiments. Overall, it may be concluded that the FE
models are able to produce accurate and consistent predictions of the test response and are thus
suitable for performing parametric studies.
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2.3. Parametric study
Based on the developed FE models, an extensive parametric study was carried out to expand
the available results over a wider range of geometries and cross-section slenderness in order to
investigate the applicability and accuracy of the codified provisions given in EN 1993-1-1 [1]
and the revised CSM for hot-rolled steel cross-sections. The parametric study included stub
column models and pure major and minor axis bending models following the basic modelling
assumptions described previously. The length of all FE models was set equal to three times the
average cross-section dimensions. The average measured material properties from the tensile
coupon tests performed in [36] on grade S355 steel were adopted in the parametric study.
The parametric study included a range of SHS and RHS, with the cross-section aspect ratios of
the latter ranging from 1.3 to 2.0. The height of the SHS was varied from 40 to 140 mm,
whereas for the RHS, the height ranged from 80 to 160 mm and the width ranged from 60 to
80 mm. For both types of cross-section (SHS and RHS), the thickness was varied between 1.3
and 5 mm and internal corner radii were taken equal to the thickness of the section. For I-
sections, the height, width and corner radii were kept constant at 300 mm, 200 mm and 15 mm,
respectively. The flange thickness was varied from 8 to 28 mm and the web thickness from 3
to 28 mm. The modelled specimens were chosen to cover the full range of local slenderness
from very compact to slender cross-sections. The obtained results are discussed in detail in the
following section.
3. Extension of the CSM to hot-rolled steel cross-sections
3.1 General concept
The CSM is a deformation (strain) based design method with two main components: (1) a base
curve defining the maximum strain εcsm that a cross-section can tolerate as a function of the
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cross-section slenderness p and (2) a material model that allows for the influence of strain
hardening. The CSM base curve relates the cross-section deformation capacity, which is
defined in a normalised form as εcsm divided by the yield strain εy, to the cross-section
slenderness p . Within the CSM, the cross-section slenderness p is defined in non-
dimensional form as the square root of the ratio of the yield stress fy to the elastic buckling
stress σcr, as given by Eq. (2). The elastic buckling stress σcr should be determined for the full
cross-section either using numerical methods, such as the finite strip software CUFSM [45], or
approximate analytical methods [12], both of which take into account the effects of plate
element interaction. Conservatively, the elastic buckling stress of the most slender individual
plate element can be used to define the slenderness of the cross-section. In the present study,
CUFSM was adopted for the determination of σcr for the full cross-section.
cryp /σfλ (2)
The CSM design base curve (Eq. (3)) and the transition point ( 68.0p ) between non-slender
and slender sections used for stainless steel sections were shown to be applicable to hot-rolled
carbon steel sections in [29]. Two upper bounds have been placed on the predicted cross-
section deformation capacity εcsm/εy; the first limit of 15 corresponds to the material ductility
requirement expressed in EN 1993-1-1 [1] and prevents excessive deformations and the second
limit of C1εu/εy, where εu is the ultimate strain of the material in tension and C1 is a coefficient
corresponding to the adopted CSM material model as described in the subsequent section,
defines a ‘cut-off’ strain to prevent over-predictions of material strength. Note that the CSM is
focussed primarily on non-slender sections where local buckling occurs after yielding, though
extension of the method to cover slender sections has recently been presented [46,47]. Note
11
also that the maximum strain limit of 15εy could be set by the designer on a project by project
basis, depending on the accepted degree of plasticity at the ultimate limit state.
0.68for 1222.0
1
0.68for ,15min≤but , 25.0
p05.1p
05.1p
p
y
u1
y
csm
6.3p
y
csm
λλλ
λε
εC
ε
ε
λ
ε
ε (3)
Owing to the existence of a yield plateau in hot-rolled steels, the elastic-linear hardening
material model employed to date in the CSM for cold-formed steel, stainless steel and
aluminium is less appropriate. The process of extending the CSM to cover the design of hot-
rolled steel cross-sections requires the selection of a suitable material model and the derivation
of simple yet accurate resistance expressions based on the adopted material model; these two
aspects are addressed in the following sub-sections.
3.2 Material model
Unlike stainless steel, aluminium and cold-formed carbon steel which exhibit predominantly
nonlinear rounded stress-strain curves, hot-rolled carbon steels have an elastic response, with
a distinctively defined yield point, followed by a yield plateau and a moderate degree of strain
hardening. Thus, a more refined material model is required for the development and application
of the CSM to hot-rolled carbon steel cross-sections. The first three stages of the quad-linear
material model proposed by Yun and Gardner [33] for hot-rolled carbon steels, as illustrated in
Fig. 9 and described in Eq. (4), is adopted herein as the CSM material model that takes account
of both the yield plateau and the strain hardening of hot-rolled steels.
12
for )(
for )(
for
for
uu1u1
u1u
uu
u
u1shshshy
shyy
y
1
1εεεCεCε
εCε
εff
εf
εCεεεεEf
εεεf
εεEε
εf
C
C
(4)
In Fig. 9, E is the Young’s modulus, fy and fu are the yield and ultimate tensile stress, εy and εu
are the strains at the yield and ultimate stresses, respectively, εsh is the strain hardening strain
where the yield plateau ends and subsequently the strain hardening initiates, C1εu represents
the strain at the intersection point of the third stage of the model and the actual stress-strain
curve, and u1Cf is the corresponding stress. Two material coefficients, C1 and C2, are used in
the material model; C1 represents the interaction point discussed previously and effectively
defines a ‘cut-off’ strain to avoid over-predictions of material strength and is also included in
the base curve (Eq. (3)); C2 is used in Eq. (5) to define the strain hardening slope Esh.
u y
sh
2 u sh
f fE
C
(5)
The predictive expressions for the required parameters of the model, εu, εsh, C1 and C2, are
given by Eqs. (6)-(9), respectively.
steels rolled-hotfor 06.0but , 16.0 u
u
yu
ε
f
fε (6)
03.00.015but , 055.01.0 sh
u
y
sh εf
fε (7)
13
sh u sh1
u
0.25( )C
(8)
sh u sh2
u
0.4( )C
(9)
3.3 CSM resistance functions for hot-rolled steels
The CSM analytical and simplified design resistance expressions for hot-rolled steel cross-
sections under compression Ncsm and bending Mcsm are derived in this section, based on the
material model described in Section 3.2. Note that only the first three stages of the quad-linear
material model are used in the CSM, since progression into the fourth stage would correspond
to very high strains and lead to overly complicated resistance functions, particularly in bending.
3.3.1 Compression resistance
The design CSM compression resistance for a hot-rolled steel cross-section Ncsm,Rd is calculated
as the product of the gross cross-section area A and the CSM limiting stress fcsm, divided by the
partial factor γM0 with a recommended value of unity for hot-rolled steel, as given by Eq. (10),
in which the fcsm may be calculated from Eq. (11) based on the first three stages of the quad-
linear material model.
M0
csmRdcsm,
AfN (10)
for )(
for
for
u1csmshshcsmshy
shcsmyy
y csmcsm
csm
CEf
f
E
f (11)
14
The post-yield strength benefit due to strain hardening can be achieved once the CSM limiting
stress εcsm obtained from the base curve Eq. (3) exceeds the strain hardening strain εsh; thereafter
the corresponding CSM limiting stress fcsm will be larger than the yield stress fy.
3.3.2 Bending resistance
For slender cross-sections, the design CSM bending resistance is given simply by the product
of the CSM limiting stress and the elastic section modulus, as given by Eq. (12).
ycsm
M0
yel
y
csm
M0
elcsmRdcsm, for
fWWfM (12)
The analytical CSM bending resistance expression for non-slender hot-rolled steel I-sections
and SHS/RHS is derived herein based on the assumption that plane sections remain plane and
normal to neutral axis in bending, and that the cross-section shape does not significantly deform
before the strain at the extreme outer-fibre εcsm is attained. The linear strain and tri-linear stress
distributions (arising from the CSM material model for hot-rolled carbon steels described in
Section 3.2) for half of a symmetric cross-section are shown in Fig. 10. The bending capacity
at a strain ratio (εcsm/εy) of unity is equal to the elastic moment Mel. With an increase in
deformation capacity (i.e. strain ratio εcsm/εy), the bending capacity is asymptotic to the plastic
moment Mpl due to the spread of plasticity until the strain ratio reaches εsh/εy, after which benefit
from strain hardening can be exploited.
The CSM cross-section resistance in bending Mcsm,Rd depends upon whether or not strain
hardening is experienced (i.e. whether or not εcsm > εsh). If εcsm ≤ εsh, the bending resistance
expressions for hot-rolled steel cross-sections are the same as those derived previously using
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the bi-linear CSM material model [29] with Esh taken equal to zero to represent the yield
plateau; the design resistance expressions are given by Eqs. (13) and (14) for major and minor
axis bending, respectively,
shcsmy
y
csm
ypl,
yel,
M0
yypl,
Rdcsm,y , for , /11 εεεε
ε
W
WfWM
α
(13)
shcsmy
y
csm
zpl,
zel,
M0
yzpl,
Rdcsm,z, for , /11 εεεε
ε
W
WfWM
α
(14)
where Wpl is the plastic section modulus, Wel is the elastic section modulus, y and z refer to the
major and minor axes, respectively, and α is a dimensionless coefficient that depends on the
type of cross-section and axis of bending, as defined in Table 3.
For the more stocky cross-sections, where εcsm > εsh, the outer fibre strain will enter into the
strain hardening regime and the CSM bending resistance can achieve Mcsm > Mpl. With
reference to Fig. 10, the bending capacity of a cross-section can now be expressed by Eq. (15),
where the appropriate section moduli Wpl,y, Ww,y and *
yw,W or Wpl,z, Ww,z and *
zw,W are used for
bending about the major and minor axes, respectively. The Ww and *
wW terms represent the
section moduli associated with the two shaded triangular stress blocks shown in Fig. 10. For
strain ratios greater than unity, the introduced modulus Ww can be approximated by Eq. (16),
the derivation of which is given in [48]. This section of the present paper is, therefore, mainly
focused on the derivation of *
wW .
16
)( ycsm*wywyplcsm ffWfWfWM (15)
α
ε
εWWW
y
csmelplw )( (16)
Substituting Eq. (16) and the CSM strength fcsm = fy + Esh(εcsm εsh) into Eq. (15) and
normalizing the equation by the plastic moment capacity Mpl = Wplfy = WplEεy gives Eq. (17).
y
ycsmsh
pl
*w
y
csm
pl
el
pl
csm 11ε
εε
E
E
W
W
ε
ε
W
W
M
Mα
(17)
The final term )( ycsm
*
w ffW in Eq. (15) represents a strain hardening moment Msh resulting
from the material strain hardening properties, which can be calculated by integration of the
triangular shaped stress block associated with the stress (fcsm – fy), for 2/* HyY (major
axis) as shown in Fig. 10, where H is the height of the cross-section and Y* is the distance from
the neutral axis to the point of first strain hardening, as given by Eq. (18).
sh
csm
*
2
HY
(18)
Considering an I-section in bending about the major axis, the derivation process for *
yw,W is
described below. The same approach can be applied to a box section (SHS or RHS), simply by
setting the wall thickness as half of the web thickness of an I-section, as shown in Fig. 11.
17
The strain hardening moment Msh,y is defined by Eq. (19), from which the introduced modulus
*yw,W can be determined using Eq. (20), where the function g(y) represents the triangular stress
distribution normalised by (fcsm – fy). The integral is evaluated from the first strain hardening
point *= Yy to
2/= Hy, and the associated area to integrate over is *Y
A .
*
*Y
*
*YYAYA
AyygffAyfffWM d)()(d)( ycsmycsm*
yw,ysh, (19)
*
*Y
YAAyygW d)(*
yw, (20)
In Fig. 12(a), the first strain hardening point lies within the web i.e. Y* ≤ hw/2. This scenario
may be alternatively expressed as:
wsh
csm ≥h
H
ε
ε (21)
In this case, the modulus *
yw,W has contributions from two parts: a trapezoidal part acting within
the flanges and a triangular part acting within the web. The modulus is calculated as:
H
h
ε
ε
ε
ε
ε
ε
H
hHtB
ε
ε
ε
εBHW w
csm
sh
1
csm
sh
2
csm
shw
2
w
csm
sh
csm
sh
2*
yw, 2112
)(21
12 (22)
When Y* lies outside the web, as shown in Fig. 12(b), the modulus *
yw,W is calculated as:
18
csm
sh
csm
sh2
*yw, 21
12 ε
ε
ε
εBHW (23)
which applies when the strain ratio is bounded by:
wsh
csm1h
H
(24)
In summary, for box and I-sections bending about their major axis, the *
yw,W modulus is defined
by Eq. (25).
1for 2112
for 2112
2112
wsh
csm
csm
sh
csm
sh2
wsh
csmw
csm
sh
1
csm
sh
2
csm
shw2
w
csm
sh
csm
sh2
*yw,
h
H
ε
ε
ε
ε
ε
εBH
h
H
ε
ε
H
h
ε
ε
ε
ε
ε
ε
H
hH)tB(
ε
ε
ε
εBH
W (25)
For box sections bending about the minor axis, the expression for *
yw,W is similar to Eq. (25),
as given in Eq. (26).
2
1for 2112
2
for 2
212
1221
12
fsh
csm
csm
sh
csm
sh2
fsh
csmf
csm
sh
1
csm
sh
2
csm
shf2
w
csm
sh
csm
sh2
*zw,
tB
B
ε
ε
ε
ε
ε
εHB
tB
B
ε
ε
B
tB
ε
ε
ε
ε
ε
ε
B
tBBh
ε
ε
ε
εHB
W (26)
19
For I-sections bent about their minor axis, the first strain hardening point is unlikely to lie
within the web, hence only the expression for*
zw,W in which the first strain hardening point lies
within the flange is considered herein, and is given by:
wsh
csm
csm
sh
csm
sh2
f*zw, for , 21
6 t
B
ε
ε
ε
ε
ε
εBtW
(27)
Substituting the expressions for the introduced modulus *
zw,W into Eq. (17) leads to the exact
analytical CSM bending equations, but these are lengthy for practical design due to the *wW
term. A simplified yet accurate approximation is therefore sought for design purposes. The
simpler design expression for *
wW is obtained based on a regression analysis of the geometric
properties of commercially available structural I-sections and SHS/RHS in [49] to determine
the dimensionless coefficient β to be used in Eq. (28); β is presented as a function of the type
of cross-section and axis of bending in Table 3.
pl
y
shcsm*w W
ε
εεW
(28)
The simpler design expression of *wW (Eq. (28)) can then be substituted into Eq. (17) to give
the simplified Mcsm equations provided as Eq. (29) and Eq. (30) for bending about the major
and minor axis, respectively.
shcsmsh
2
y
shcsm
y
csm
ypl,
yel,
M0
yypl,
Rdcsm,y, for , /11 εεE
E
ε
εεβ
ε
ε
W
W
γ
fWM
α
(29)
20
shcsmsh
2
y
shcsm
y
csm
zpl,
zel,
M0
yzpl,
Rdcsm,z, for , /11 εεE
E
ε
εεβ
ε
ε
W
W
γ
fWM
α
(30)
4. Assessment of proposals and reliability analysis
Ultimate capacities from the numerical parametric studies, together with the experimental
results summarised in Tables 1 and 2, on stub columns (Nu,FE(test)) and beams (Mu,FE(test)),
respectively, have been used to assess the accuracy of the resistance predictions of the CSM
(Nu,csm and Mu,csm). For comparison purposes, the codified design provisions of EN 1993-1-1
[1] (Nu,EC3 and Mu,EC3) are also examined. Note that the comparisons were performed using the
measured geometric and material properties of the tested and modelled cross-sections, with all
partial safety factors set to unity. Cross-sections with 68.015.0 p , which cover most of
the commercially available non-slender hot-finished I-sections and SHS/RHS, were the focus
of the present study. The comparisons are presented in Figs 13-17, in which the experimentally
and numerically obtained ultimate capacities have been normalised by the yield load Ny and
the plastic moment capacity Mpl for cross-sections in compression and in bending, respectively,
and are plotted against the cross-section slenderness p . Curves indicating design capacities
from EN 1993-1-1 [1] and the CSM have also been plotted together with the experimental and
numerical data in Figs 13-17. Since the CSM curve depends on the shape of the cross-sections
and the material properties, a curve based on the average geometric and material properties
used in the parametric studies has been depicted for illustration purposes. The EN 1993-1-1 [1]
resistance curves have been positioned based on the slenderness limits corresponding to the
Class 2 limiting width-to-thickness ratio given in Table 5.2 of EN 1993-1-1, applied on an
element-by-element basis [42] i.e. for internal compression elements, the Class 2 width-to-
thickness ratio of 38ε (ε is the material factor yf/235 ) corresponds to 67.0p (see
21
Fig. 15), and for outstand elements in compression, the Class 2 width-to-thickness ratio of 10ε
corresponds to 54.0p (see Figs 16 and 17). The mean slenderness limits corresponding to
the Class 2 width-to-thickness ratio in EN 1993-1-1 based on the elastic buckling stress of the
cross-sections considered in this study, determined using CUFSM [45], are also depicted in
Figs 15-17. In Figs 15 and 16, the slenderness limits determined on an element-by-element
basis (i.e. based on EN 1993-1-1) are similar to the values determined based on buckling of the
full cross-section (i.e. using σcr from CUFSM, with some influence from element interaction.
However, for I-sections in bending about the minor axis, the web, which is at a low level of
stress, and the outstand flanges on the tension side, which experience stabilising tensile
stresses, remain unbuckled and thus provide significant assistance to the flange plates in
compression. In this instance, the boundary conditions on the unloaded edges of the
compressed outstands are approaching fixed-free. However, in EN 1993-1-1, the outstand
flanges of I-sections are treated, ignoring element interaction, as pinned-free, resulting in
significantly higher slenderness values when compared to the mean CUFSM value, as
highlighted in Fig. 17. The key results from the comparisons are provided in Table 4 for the
cross-sections in compression and Table 5 for the cross-sections in bending.
4.1 Assessment of CSM for hot-rolled steel cross-sections in compression
The compressive strengths predicted by EN 1993-1-1 [1] and the CSM have been compared
with experimental and numerical results for both SHS/RHS and I-sections, as shown in Table
4 and Figs 13 and 14. The comparisons demonstrate that the very stocky hot-rolled steel cross-
sections, i.e. 35.0p , have the potential to exceed their yield load substantially. It can be
seen from Figs 13 and 14 that the CSM provides the same predictions for cross-section
resistance in the local slenderness range 68.035.0 p due to the existence of the yield
22
plateau in the adopted CSM material model, while for stockier cross-sections ( 35.0p ),
where the predicted failure strain εcsm extends into the strain hardening region, the CSM yields
improved resistance predictions over EN 1993-1-1 [1]. As shown in Table 4, for the more
stocky sections with 35.0p , the mean values of FE (or test)-to-predicted ultimate loads for
the EN 1993-1-1 [1] (Nu,FE(test)/Nu,EC3) and the CSM (Nu,FE(test)/Nu,csm) are 1.28 and 1.22 for the
I-sections and 1.20 and 1.16 for the SHS/RHS, with the corresponding COV of 0.055 and 0.049
for the I-sections and 0.089 and 0.078 for the SHS/RHS, respectively. The CSM predictions
remain conservative for the stocky cross-sections due primarily to the imposed strain limit of
εcsm = 15εy. This limit could be adjusted if a designer wished to impose higher or lower strain
limits on a given project.
4.2 Assessment of CSM for hot-rolled steel cross-sections in bending
Comparisons between the test and FE capacities of cross-sections in bending and the CSM and
EN 1993-1-1 [1] resistance predictions are shown in Figs 15-17, while the key results of the
comparisons are reported in Table 5. Based on the cross-section types and axis of bending, the
data have been classified into 3 groups: I-sections bent about their major axis, I-sections bent
about their minor axis and SHS/RHS bent about either axis. For the more stocky sections with
35.0p , the EN 1993-1-1 [1] predictions are unduly conservative, especially for I-sections,
with mean values of Mu,FE(test)/Mu,EC3 equal to 1.29 and 1.30 for I-sections bent about their major
and minor axis, respectively. The conservatism is less pronounced in the CSM predictions,
except for I-sections bent about their minor axis, where high strains are needed to reach the
plastic bending moment Mpl due to the high cross-section shape factor (i.e. the ratio of plastic-
to-elastic section modulus Wpl/Wel); here, the limited strain hardening is largely compensating
for the more gradual plastification of the section to enable the attainment of Mpl, rather than
bringing resistance benefits beyond Mpl. This point highlights one of the additional benefits of
23
the CSM, which is knowledge of the strains required to attain a given capacity; this information
is not available in traditional strength-based design. Note that if the maximum CSM strain ratio
limit were to be relaxed to 20 or 25, further exploitation of strain hardening would be possible
and a closer match with the ultimate test and FE capacities is achieved, as indicated in Figs 13-
17.
For slenderness values 35.0p , the CSM is shown to offer more accurate ultimate capacity
predictions and far fewer on the unsafe side towards the slender end of the Class 2 range by
allowing systematically for the increasing bending resistance with increasing deformation
capacity (i.e. strain ratio εcsm/εy) due to the spread of plasticity. Specifically, for Class 3 cross-
sections, the CSM yields significantly more accurate than EN 1993-1-1 due to the allowance
for the partial plastification, with mean values of Mu,FE(test)/Mu,csm in this slenderness range of
1.03, 1.09 and 1.03 for I-sections in major axis bending, I-sections in minor axis bending and
SHS/RHS in bending about either axis, respectively, all of which are lower (i.e. closer to unity
and hence more accurate) than the corresponding values of Mu,FE(test)/Mu,EC3. Moreover, the
nature of the cross-section classification system in EN 1993-1-1 [1] results in a sharp step in
the predicted resistance at a discrete slenderness limit; this step is more pronounced for the I-
sections bent about their minor axis, as shown in Fig. 16, due to the higher ratio of Wpl/Wel,
while this discontinuity is eliminated in the CSM approach.
4.3 Reliability analysis
A statistical analysis is presented in this section to assess the reliability level of the proposed
CSM resistance expressions for hot-rolled steel cross-sections according to Annex D of EN
1990 [50]. In the reliability analysis, the material over-strength, i.e. the mean-to-nominal yield
strength ratio fy,mean/fy,norm, was taken as 1.16 and the COVs of material strength and geometric
24
properties were taken as 0.05 and 0.03, respectively, in accordance with the values
recommended by Byfield and Nethercot [51]. The key statistical parameters for the proposed
CSM expressions are summarized in Tables 6 and 7, including the number of tests and FE
simulations, the design (ultimate limit state) fractile factor kd,n, the average ratio of test (or FE)-
to-model resistance based on a least square fit to all data b, the coefficient of variation of the
tests and FE simulations relative to the resistance model Vδ, the combined coefficient of
variation incorporating both model and basic variable uncertainties Vr and the required partial
safety factor γM0. The results of the reliability analysis based on the EN 1993-1-1 [1] predictions
are also reported in Tables 6 and 7 for comparison purposes. For I-sections in compression and
in bending, the obtained partial safety factors γM0 were found to be less than or very close to
unity, which is the target value in EN 1993-1-1, for both EN 1993-1-1 and the CSM, as shown
in Tables 1 and 2. The CSM is therefore deemed to be reliable for the design of hot-rolled steel
I-sections. For SHS/RHS, the obtained partial safety factors γM0 for the CSM are lower than
the corresponding values for EN 1993-1-1 [1], with CSM partial safety factors γM0 of 1.16 for
compression and 1.11 for bending and EN 1993-1-1 partial safety factors γM0 of 1.20 for
compression and 1.18 for bending. Therefore, although the CSM partial safety factors γM0 are
slightly higher than the target value of 1.0, the CSM design expressions are deemed to be
acceptable on the basis of historical precedent [50,52].
5. Conclusions
The continuous strength method (CSM) has been extended herein to cover the design of hot-
rolled steel cross-sections, including I-sections and SHS/RHS. The developments of the
method and the derivation of the resistance functions has been described. Numerical models
were created and thoroughly validated against available test results for hot-rolled steel cross-
sections under compression and bending. Upon validation of the FE models, a comprehensive
25
parametric study was conducted to generate further numerical data on hot-rolled steel cross-
sections under different loading cases. The experimental and numerical data were then used to
evaluate the accuracy of the provisions of EN 1993-1-1 [1] and the extended CSM for the
structural design of hot-rolled steel cross-sections.
A refined quad-linear material model considering both the yield plateau and the strain
hardening of hot-rolled steels was adopted as the CSM material model, and the corresponding
cross-section resistance expressions for axial compression and bending were presented.
Comparisons with experimental and numerical data revealed that the CSM generally provides
more accurate and consistent predictions than the existing design provisions, especially for
very stocky cross-sections and for Class 3 sections in bending. Reliability analysis of the EN
1993-1-1 and the CSM design provisions was conducted against test and FE data according to
EN 1990 [50]; it was found that the CSM satisfies the reliability requirements for I-sections
(i.e. the required value of γM0 is less than unity) and requires lower partial safety factors γM0
than the corresponding values of EN 1993-1-1 [1] for SHS/RHS. Further work is currently
underway to extend the CSM to cover combined loading and high strength steels.
Acknowledgements
The financial support provided by the China Scholarship Council (CSC) for the first author’s
PhD study at Imperial College London is gratefully acknowledged.
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31
Tables:
Table 1 Comparison of stub column test results with FE results for varying imperfection amplitudes.
Reference Steel
grade Specimen
Cross-
section
Nu,FE/Nu,test
Imperfection amplitude
a/400 a/200 a/100 D-W model
Gardner et al. [4]
S355 SHS 100×100×4-HR1 SHS 1.07 1.06 1.06 1.08
S355 SHS 100×100×4-HR2 SHS 1.05 1.04 0.99 1.08
S355 SHS 60×60×3-HR1 SHS 0.95 0.94 0.93 0.97
S355 SHS 60×60×3-HR2 SHS 0.93 0.92 0.91 0.95
S355 RHS 60×40×4-HR1 RHS 0.97 0.95 0.93 1.04
S355 RHS 60×40×4-HR2 RHS 0.97 0.94 0.92 1.03
S355 SHS 40×40×4-HR1 SHS 0.94 0.89 0.82 0.98
S355 SHS 40×40×4-HR2 SHS 0.93 0.89 0.82 0.97
S355 SHS 40×40×3-HR1 SHS 0.87 0.85 0.84 0.93
S355 SHS 40×40×3-HR2 SHS 0.89 0.87 0.85 0.94
Nseir [35]
S355 LC1_HF_250×150×5 RHS 1.05 1.01 0.95 1.11
S355 LC1_HF_200×100×5 RHS 1.06 1.04 1.01 1.06
S355 LC1_HF_200×200×5 SHS 1.04 0.98 0.89 1.12
S355 LC1_HF_200×200×6.3 SHS 1.04 1.04 1.03 1.04
S355 Stub_HF_250×150×5 RHS 1.08 1.04 0.99 1.16
S355 Stub_HF_200×100×5 RHS 1.06 1.02 0.97 1.07
S355 Stub_HF_200×200×5 SHS 1.09 1.04 0.98 1.11
S355 Stub_HF_200×200×6.3 SHS 0.97 0.94 0.86 1.02
Foster et al. [37]
- 305×127×48-SC1 I-section 0.97 0.96 0.93 0.98
- 305×127×48-SC2 I-section 0.99 0.98 0.95 1.00
- 305×165×40-SC1 I-section 1.02 1.01 0.98 1.06
- 305×165×40-SC2 I-section 0.99 0.97 0.94 1.02
Yun et al. [36] S235 P7 HEB160 I-section 0.89 0.87 0.85 0.94
S355 C7 HEB160 I-section 0.90 0.87 0.86 0.91
Mean 0.99 0.96 0.93 1.02
COV 0.068 0.069 0.073 0.066
32
Table 2 Comparison of bending test results with FE results for varying imperfection amplitudes.
Reference Steel grade Specimen Cross-
section
Bending
configuration
Mu,FE/Mu,test
Imperfection amplitude
a/400 a/200 a/100 D-W model a/200
(Pure bending model)
Byfield and
Nethercot [6]
FE430A Y1 203×102×23UB I-section 4-point 0.91 0.88 0.84 0.92 0.89
FE430A Y2 203×102×23UB I-section 4-point 0.91 0.88 0.85 0.92 0.90
FE430A Y3 203×102×23UB I-section 4-point 0.95 0.93 0.90 0.96 0.95
FE430A Y4 203×102×23UB I-section 4-point 0.90 0.87 0.84 0.92 0.89
FE430A Y5 203×102×23UB I-section 4-point 0.94 0.90 0.87 0.95 0.92
FE430A Y6 203×102×23UB I-section 4-point 0.95 0.92 0.89 0.96 0.94
Wang et al. [38]
S460 SHS 50×50×5 SHS 3-point 1.00 1.00 1.00 1.00 -
S460 SHS 50×50×4 SHS 3-point 1.06 1.06 1.06 1.06 -
S460 SHS 100×100×5 SHS 3-point 0.98 0.98 0.98 0.98 -
S460 SHS 90×90×3.6 SHS 3-point 1.01 1.01 1.01 1.01 -
S460 RHS 100×50×6.3 RHS 3-point 0.96 0.96 0.96 0.96 -
S460 RHS 100×50×4.5 RHS 3-point 1.11 1.11 1.11 1.11 -
S460 SHS 50×50×5 SHS 4-point 0.99 0.99 0.99 0.99 1.00
S460 SHS 50×50×4 SHS 4-point 0.97 0.97 0.97 0.97 0.98
S460 SHS 100×100×5 SHS 4-point 0.88 0.88 0.86 0.88 0.88
S460 SHS 90×90×3.6 SHS 4-point 0.92 0.91 0.88 0.92 0.92
S460 RHS 100×50×6.3 RHS 4-point 0.87 0.87 0.87 0.87 0.87
S460 RHS 100×50×4.5 RHS 4-point 0.91 0.90 0.90 0.91 0.91
0.96 0.94 0.93 0.96 0.92
0.065 0.074 0.084 0.062 0.044
33
Table 3 CSM coefficients α and β for use in bending resistance functions.
α β
Axis of bending Major Minor Major Minor
I-section 2 1.2 0.1 0.05
SHS/RHS 2 2 0.1 0.1
Table 4 Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test results for cross-sections in
compression.
Cross-
section
type
35.0p λ All ( 68.0p λ )
No. of FE
(tests)
Nu,FE(test)/Nu,EC3 Nu,FE(test)/Nu,csm No. of FE
(tests)
Nu,FE(test)/Nu,EC3 Nu,FE(test)/Nu,csm
Mean (COV) Mean (COV) Mean (COV) Mean (COV)
I-sections 82 (0) 1.28 (0.055) 1.22 (0.049) 140 (4) 1.17 (0.117) 1.14 (0.098)
SHS/RHS 26 (6) 1.20 (0.089) 1.16 (0.078) 159 (10) 1.03 (0.097) 1.02 (0.080)
34
Table 5 Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test results for cross-sections in bending.
Cross-section - axis of
bending
35.0p Class 3 All ( 68.0p )
No. of FE
(tests)
Mu,FE(test)/Mu,EC3 Mu,FE(test)/Mu,csm No. of FE
(tests)
Mu,FE(test)/Mu,EC3 Mu,FE(test)/Mu,csm No. of FE
(tests)
Mu,FE(test)/Mu,EC3 Mu,FE(test)/Mu,csm
Mean (COV) Mean (COV) Mean (COV) Mean (COV) Mean (COV) Mean (COV)
I-sections - major 180 (6) 1.29 (0.033) 1.26 (0.032) 27 (0) 1.10 (0.014) 1.03 (0.016) 265 (6) 1.22 (0.088) 1.20 (0.088)
I-sections - minor 98 (0) 1.30 (0.054) 1.30 (0.053) 41 (0) 1.48 (0.043) 1.09 (0.049) 147 (0) 1.34 (0.090) 1.23 (0.096)
SHS/RHS - both axes 82 (6) 1.15 (0.069) 1.13 (0.064) 35 (0) 1.07 (0.012) 1.03 (0.023) 309 (9) 1.03 (0.091) 1.03 (0.071)
35
Table 6 Summary of reliability analysis results for cross-sections in compression.
Design approach Cross-section No. of tests and simulations kd,n b Vδ Vr γM0
EN 1993-1-1 I-sections 144 3.159 1.231 0.119 0.132 1.05
SHS/RHS 169 3.149 0.996 0.088 0.106 1.20
CSM I-sections 144 3.159 1.193 0.099 0.115 1.03
SHS/RHS 169 3.149 0.994 0.074 0.094 1.16
Table 7 Summary of reliability analysis results for cross-sections in bending.
Design
approach
Cross-section - axis
of bending
No. of tests and
simulations kd,n b Vδ Vr γM0
EN 1993-1-1
I-sections - major 271 3.126 1.270 0.092 0.109 0.95
I-sections - minor 147 3.158 1.318 0.091 0.108 0.91
SHS/RHS - both axes 318 3.121 1.007 0.087 0.105 1.18
CSM
I-sections - major 271 3.126 1.250 0.092 0.109 0.96
I-sections - minor 147 3.158 1.300 0.115 0.099 0.94
SHS/RHS - both axes 318 3.121 1.020 0.068 0.090 1.11
36
Figures:
Fig. 1. Representation of the fillet zones of I-sections in FE models.
(a) SHS/RHS stub column. (b) I-section stub column.
(c) SHS/RHS in 4-point bending.
(d) I-section in 4-point bending.
Fig. 2. Typical elastic buckling mode shapes adopted as initial local imperfections in FE models.
Nodes of flange
Nodes of web
Fillet zone
Nodes of flange
Nodes of web
Increased web thickness
37
(a) H/B ≤ 1.2.
(b) H/B > 1.2.
Fig. 3. Residual stress patterns applied to FE models for I-sections.
0.5fy*
0.5fy*0.5fy
*
0.5fy*
0.5fy*
0.5fy*
H
B
ritf
tw
0.3fy*
0.3fy* 0.3fy
*
0.3fy*
0.3fy*
0.3fy*
H
B
ri
tf
tw
38
(a) Residual stress pattern presented in DIN [44].
(b) Simplified residual stress pattern.
Fig. 4. Residual stress patterns for SHS/RHS.
0.5fy* 0.5fy
*
0.2fy*
0.2fy*
0.5fy*
0.5fy*
B
H
ri
t
0.5fy* 0.5fy
*
B
H
ri
t
0.5fy*
0.5fy*
σf=2fy*(ri+t/2)sin(π/8)/(B-2ri-2t)
σf
σw σw=2fy*(ri+t/2)sin(π/8)/(H-2ri-2t)
39
Fig. 5. Comparison of experimental and numerical load-end shortening curves for C7 HEB160 [36].
Fig. 6. Comparison of experimental and numerical load-end shortening curves for Stub_HF_200×200×5 [35].
Fig. 7. Comparison of experimental and numerical moment-end rotation curves for Y6 203×102×23UB [6].
0
500
1000
1500
2000
2500
3000
0 5 10 15 20 25
N(k
N)
δ (mm)
Test
FE
0
500
1000
1500
2000
2500
0 1 2 3 4 5 6 7 8
N(k
N)
δ (mm)
Test
FE
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6 7 8 9
M(k
Nm
)
θ (degree)
TestFE
40
Fig. 8. Comparison of experimental and numerical normalized moment-curvature curves for SHS 50×50×5 [38].
Fig. 9. Typical measured stress-strain curve for hot-rolled carbon steel and proposed quad-linear material model.
Fig. 10. Strain and stress distributions for half of an I-section or SHS/RHS in bending.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5 6 7 8
M/M
pl
κ/κpl
TestFE
εcsm fyfcsm-fy
εsh
εy
Y
H/2
(B
/2)
Y*
Neutral axis
*
wW
wW
y
41
Fig. 11. Cross-section geometry for the CSM analytical bending expressions (I-section and SHS/RHS).
(a) Y* within the web.
(b) Y* within the flange.
Fig. 12. Geometry for the derivation of *
yw,W for I-sections.
B
H
y-y
B
H
z-z z-z
hw
hw
tw=t tw=t/2
tf
tf
B
H
z-z
hw
tw
tf
Y*
y-yNeutral axis
fy fcsm-fy
*
yw,W
fcsm-fyB
H hw
Neutral axis y-y
*
yw,W
Y*
tw
tf
fyz-z
42
Fig. 13. Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test data for SHS/RHS in
compression.
Fig. 14. Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test data for I-sections in
compression.
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Nu,F
E(t
est)/N
y
Cross-section slenderness λp
TestFECSM strain ratio limit to 15CSM strain ratio limit to 20EN 1993-1-1
εcsm/εy=20
εcsm/εy=15
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Nu
,FE
(test
)/N
y
Cross-section slenderness λp
TestFECSM strain ratio limit to 15CSM strain ratio limit to 20EN 1993-1-1
εcsm/εy=20
εcsm/εy=15
43
Fig. 15. Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test data for SHS/RHS in
bending.
Fig. 16. Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test data for I-sections under
major axis bending.
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mu
,FE
(test
)/M
pl
Cross-section slenderness λp
TestFECSM strain ratio limit to 15CSM strain ratio limit to 20EN 1993-1-1
εcsm/εy=20
εcsm/εy=15
Class 2 limit utilising CUFEM (λp ≈ 0.65)
Class 2 limit based on EN 1993-1-1 (λp ≈ 0.67)
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mu
,FE
(tes
t)/M
pl
Cross-section slenderness λp
TestFECSM strain ratio limit to 15CSM strain ratio limit to 20EN 1993-1-1
εcsm/εy=20
εcsm/εy=15
Class 2 limit utilising CUFSM (λp ≈ 0.51)
Class 2 limit based on EN 1993-1-1 (λp ≈ 0.54)
44
Fig. 17. Comparison of CSM and EN 1993-1-1 resistance predictions with FE/test data for I-sections under
minor axis bending.
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mu
,FE
(tes
t)/M
pl
Cross-section slenderness λp
FECSM strain ratio limit to 15CSM strain ratio limit to 20CSM strain ratio limit to 25EN 1993-1-1
εcsm/εy=25
εcsm/εy=20
εcsm/εy=15
Class 2 limit utilising CUFSM (λp ≈ 0.39)
Class 2 limit based on EN 1993-1-1 (λp ≈ 0.54)