Flexural-Torsional Buckling Assessment of Steel Beam-Columns Through A Stiffness Reduction Method
Flexural-Torsional Buckling Assessment of Steel Beam-Columns Through A Stiffness Reduction Method
Flexural-Torsional Buckling Assessment of Steel Beam-Columns Through A Stiffness Reduction Method
Abstract
In this paper, a stiffness reduction method for the flexural-torsional buckling assessment of
steel beam-columns subjected to major axis bending and axial compression is presented.
The proposed method is applied by reducing the Young’s E and shear G moduli through
the developed stiffness reduction functions and performing Linear Buckling Analysis. To
account for second-order forces induced prior to buckling, the in-plane (in the plane of
bending) and out-of-plane analyses of a member are separated and stiffness reduction for
the out-of-plane instability assessment is applied on the basis of member forces determined
from the in-plane analysis. Since the developed stiffness reduction functions fully take into
account the detrimental influence of imperfections and spread of plasticity, the proposed
method does not require the use of member design equations, thus leading to practical
design. For the purpose of verifying this approach, the strength predictions determined
through the proposed stiffness reduction method are compared against those obtained from
nonlinear finite element modelling for a large number of regular, irregular, single and multi-
span beam-columns.
Keywords: Stiffness reduction; flexural-torsional buckling; steel beam-columns; inelastic
buckling
1. Introduction
The elastic flexural-torsional buckling capacities of steel beam-columns are eroded by
the effects of imperfections, residual stresses and the onset and spread of plasticity. In
current steel design specifications [1–3], this is traditionally taken into account by reduc-
ing cross-sectional resistances through buckling reduction factors, separating the individual
components of loading and resistance (i.e. flexural buckling under axial load and lateral-
torsional buckling under major axis bending) and considering their interdependency through
∗
Corresponding author
Email addresses: merih.kucukler10@imperial.ac.uk (Merih Kucukler),
leroy.gardner@imperial.ac.uk (Leroy Gardner), l.macorini@imperial.ac.uk (Lorenzo Macorini)
Load Load
My,Ed
Elastic
buckling Elastic Elastic
load buckling buckling
x
Inelastic
buckling
uz uy z load Inelastic Inelastic
buckling buckling
My,Ed
In-plane Out-of-plane
deformation - u z deformation - u y , x
NEd
y
(a) Perfectly straight steel (b) Load-displacement response
beam-column
stiffness reduction method for regular, irregular, single and multi-span beam-columns. The
accuracy of the proposed method is verified by comparing its results against those obtained
through Geometrically and Materially Nonlinear Analysis with Imperfections (GMNIA) us-
ing finite element modelling. Moreover, the proposed approach is also compared against
the traditional beam-column design methods given in EN 1993-1-1 [1], and its qualities in
comparison to these methods are shown. In this paper, the application of the proposed
stiffness reduction method is investigated for hot-rolled steel members with Class 1 and 2
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT
cross-sections [1].
fu
fy Esh
E
Strain, ε
εy εsh εu
Figure 3: Residual stress patterns applied to finite element models (+ve=tension and -ve=compression)
whereby localised failure at the end supports was also prevented. The nodes within the end
sections of the members were constrained to a reference point located at the centroid of
the cross-section where the boundary conditions were applied. The accuracy of this mod-
elling technique was verified by comparing the elastic buckling moments obtained through
the finite element models subjected to uniform bending with those obtained through the
analytical formulae provided by Trahair [8] for fork-end supported members.
NEd
U, Magnitude
1.0
0.8
0.6
0.4
0.2
0.0
My,Ed z
x
NEd y
Figure 4: Adoption of the lowest global buckling mode as the geometrical imperfection for a fork-end
supported beam-column under axial compression plus uniform bending
times of the actual length L [23]. In the finite element models, these restraint conditions were
replicated using elastic rotational springs at the supports whose stiffness was specified to
provide this same effective length value. Since the shapes and magnitudes of the geometrical
imperfections of the specimens were not provided by Van Kuren and Galambos [23], they
were assumed as a half-sine wave in shape and L/1000 in magnitude in both the in-plane
and out-of-plane directions. The patterns and magnitudes of the residual stresses of the
cross-sections used in the experiments were investigated in separate studies [25, 26], whose
recommendations were used herein. The loading sequence, whereby the axial compression
was applied first and kept constant while the bending moment was increased up to the
collapse, and material properties reported by Van Kuren and Galambos [23] were adopted
in the finite element models. The normalised moment-rotation paths of the specimens T23,
T26, T31 and T32 observed in the experiments and those determined through the finite
element models are provided in Fig. 5. The geometrical properties and loading conditions of
the specimens, which had W 100×100×19.3 steel cross-sections, are also shown in the figure,
where My,Ed is the applied major axis bending moment, θ is the end rotation, Npl = Afy
is the yield load, in which fy is the yield stress and A is the cross-sectional area, and
My,pl = Wpl,y fy is the major axis plastic bending moment resistance, in which Wpl,y is the
major axis plastic section modulus. Fig. 5 shows that the agreement between the normalised
moment-rotation paths obtained through the GMNIA of the finite element models and
those of the experiments is good, which indicates that the finite element models are able to
replicate the physical response of steel beam-columns influenced by out-of-plane instability
effects. Table 1 also shows the comparison between the ultimate strength values predicted
6
1.0 0.114Νpl 0.122Νpl 0.122Νpl 0.122Νpl
Μy,Ed Μy,Ed
T23
0.8
T31
y,pl
θ
θ
y,Ed
θ
0.4 T32
M
Μy,Ed Μy,Ed
0.114Νpl 0.122Νpl
Μy,Ed Μy,Ed
0.2
T23 T26 0.122Νpl 0.122Νpl
Van Kuren & Galambos (1961)
FE - Shell T31 T32
0
0 0.05 0.1 0.15
End rotation - θ (rad) z
y
W 100x100x19.3
Figure 5: Comparison between the normalised moment end-deformation paths of the FE models and those
from experiments
by the shell finite element models against those obtained from the experiments for eleven
specimens, where λy is the slenderness of the specimen determined by dividing its length L
to the radius of gyration of its cross-section about the major axis iy , (i.e. λ = L/iy ), NEd
is the applied axial load, Mult,exp and Mult,F E are the ultimate bending moment resistances
obtained from the experiment and the finite element model respectively, and ζ is the ratio
of the ultimate bending moment resistance obtained from the finite element model to that
observed in the experiment, (i.e. ζ = Mult,F E /Mult,exp ). ζav and ζcov are the average and
coefficient of variation of ζ values, and ζmax and ζmin are the maximum and minimum of
ζ values respectively. The three types of loading conditions of the specimens (i.e. a, b
and c) are illustrated in Fig. 6. It should be noted that the specimens of Van Kuren and
Galambos [23] not influenced by instability effects (i.e. those exhibiting significant amounts
of strain hardening) and those subjected to minor axis bending or pure compression were
not considered herein. As can be seen from Table 1, the agreement between the ultimate
bending moment strengths determined through the finite element models and those observed
in the experiments is generally good. The discrepancies in the maximum strength values
may result from the differences between the actual shapes and magnitudes of the geometrical
imperfections of the specimens which were not reported by Van Kuren and Galambos [23]
and those assumed in the finite element models. Additional validation studies of the shell
finite element modelling approach adopted in this paper against different experiments from
the literature can be found in Kucukler et al. [27] and Kucukler [28].
7
Table 1: Comparison of the ultimate strength values determined through the shell finite element models
against those obtained from the beam-column experiments of Van Kuren and Galambos [23]
a a
My,Ed z
My,Ed λy = L / iy y
a-a
Load case a Load case b Load case c
Figure 6: Considered loading cases in Van Kuren and Galambos [23] beam-column experiments
Aspect ratio αz αq
LT
W
h/b ≤ 1.2 0.49 0.22 Wel,y
q el,z
W
h/b > 1.2 0.34 0.17 Wel,y
el,z
by Kucukler et al. [16, 27]. In the following section, these functions will be utilised in the
derivation of a stiffness reduction function for the flexural-torsional buckling assessment of
beam-columns.
4ψ 2
τN z = h q i2 but τN ≤ 1
NEd /Npl −1
αz2 NEd /Npl 1 + 1 − 4ψ α2 NEd /Npl
z
NEd NEd
where ψ = 1 + 0.2αz − (1)
Npl Npl
It is worth noting that the proposed stiffness reduction function yields the same flexural
buckling strengths as those determined through the EN 1993-1-1 [1] column buckling curves
and fully accounts for the development of plasticity and the effects of residual stresses and
geometrical imperfections on the flexural buckling strength.
2
4ψLT
τLT = h q i2 but τLT ≤ 1
2 M /M −1
κ2 αLT My,Ed /My,pl 1 + 1 − 3.2ψLT κ2 α2y,EdMy,Edy,pl
/My,pl
LT
My,Ed My,Ed
where ψLT = 1 + 0.2καLT − (2)
My,pl My,pl
W /A
κ= r pl,y (3)
2
GIt GIt
8My,pl
+ 8My,pl
+ Iw /Iz
1.3
NEd My,Ed
1/αult,c = + (4)
Npl My,pl
s
NEd NEd
Mcr,N,i = τN,LT Mcr 1− 1− (5)
τN,LT Ncr,z τN,LT Ncr,T
s
π 2 EIz π 2 EIw
Mcr = GIt + (6)
L2 L2
π 2 EIw
A
Ncr,T = GIt + (7)
Iy + Iz L2
π 2 EIz
Ncr,z = (8)
L2
For a beam-column to be deemed adequate, the out-of-plane buckling load factor αcr,op
must be greater than or equal to 1.0 as shown in eq. (9). Note that after reducing the
Young’s E and shear G moduli through τN,LT , Linear Buckling Analysis (LBA-SR) directly
provides αcr,op values in conjunction with the corresponding buckling modes, where the
buckling mode with the lowest αcr,op controls the design.
Mcr,N,i
αcr,op = ≥ 1.0
My,Ed
τN,LT Ncr,z τN,LT Ncr,T
αcr,op = ≥ 1.0 ; αcr,op = ≥ 1.0 (9)
NEd NEd
In the stiffness reduction functions for flexural buckling τN given by eq. (1) and for
LTB τLT by eq. (2), members subjected to pure compression alone or major axis bending
moment alone are considered. Thus, the full cross-section capacities Npl and My,pl are used
in the functions. However, for members subjected to combined major axis bending and axial
compression, the combined effect of the actions on the degree of stiffness reduction should be
considered, as recommended by Trahair and Hancock [12]. Considering combined loading,
the reduced bending moment My,pl,r and axial load Npl,r resistances can be determined using
eq. (10) and eq. (11) respectively, which were determined by re-arrangement of the ultimate
cross-section interaction equation given in eq. (4). It should be noted that though the
11
ultimate cross-section resistance equation proposed by Duan and Chen [29] is used herein,
alternative equations such as that given in EN 1993-1-1 [1] can be employed to determine
the reduced cross-section resistances.
1/1.3
My,Ed
Npl,r = Npl 1− (10)
My,pl
" 1.3 #
NEd
My,pl,r = My,pl 1 − (11)
Npl
The ratios between the applied loads and corresponding reduced section capacities are
used to modify the stiffness reduction functions for flexural buckling τN z and LTB τLT as
shown in eq. (12) and eq. (13). The only difference between the stiffness reduction functions
given in eq. (1) and eq. (2) and those presented in eq. (12) and eq. (13) is the use of the
reduced section capacities. It is worth noting that the stiffness reduction functions given by
eq. (1) and eq. (2) were originally derived from the Perry-Robertson equations that take
into account the elastic and first yield mechanical flexural and lateral-torsional buckling
response of geometrically imperfect steel members but involve imperfection factors αz and
αLT calibrated to the GMNIA results for the consideration of residual stresses and the spread
of plasticity [16, 27].
4ψ 2
τN z,R = h q i2 but τN z,R ≤ 1
NEd /Npl,r −1
αz2 NEd /Npl,r 1 + 1 − 4ψ α2 NEd /Npl,r
z
NEd NEd
ψ = 1 + 0.2αz − (12)
Npl,r Npl,r
2
4ψLT
τLT,R = h q i2 but τLT,R ≤ 1.0
2 2 My,Ed /My,pl,r −1
κ αLT My,Ed /My,pl,r 1 + 1 − 3.2ψLT κ2 α2 My,Ed /My,pl,r
LT
My,Ed My,Ed
where ψLT = 1 + 0.2καLT − (13)
My,pl,r My,pl,r
It is proposed herein that the expression for the stiffness reduction function for the
flexural-torsional buckling assessment of beam-columns τN,LT is given by the multiplication
of the modified stiffness reduction functions τN z,R and τLT,R . The proposed expression for
τN,LT is therefore as given by eq. (14). Note that τN,LT degenerates into the stiffness
reduction functions developed for pure loading cases: when the bending moment is zero
τN,LT = τN z and when the axial force is zero τN,LT = τLT .
4.3.1. Incorporation of moment gradient factors Cm,LT into the stiffness reduction functions
This subsection describes the incorporation of moment gradient factors Cm,LT into the
stiffness reduction functions for the consideration of the influence of moment gradient on
13
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT
0.5My,Ed NEd
0.5My,Ed
0.5My,Ed
Figure 7: Example application of the stiffness reduction method to steel beam-columns subjected to major
axis bending and axial compression
the development of plasticity. The moment gradient factors are incorporated into the com-
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT
ponents of the stiffness reduction function τN,LT associated with bending as shown in eq.
(15) and eq. (16), where My,Ed is the absolute maximum value of the bending moment (i.e.
first + second-order) along the laterally unrestrained length of the beam-column. Similar to
Kucukler et al. [27], this study also recommends the determination of different Cm,LT and
τN,LT for each laterally unrestrained segment of the beam-column to accurately locate the
failing segment.
1/1.3
Cm,LT My,Ed
Npl,r = Npl 1 − (15)
My,pl
2
4ψLT
τLT,R = h q i2
2 Cm,LT My,Ed /My,pl,r −1
κ2 αLT Cm,LT My,Ed /My,pl,r 1 + 1 − 3.2ψLT κ2 α2 Cm,LT My,Ed /My,pl,r
LT
q
2
7My,Ed + 5MA2 + 8MB2 + 5MC2
Cm,LT = ≥ 0.7 (17)
5My,Ed
It should be emphasised that the moment gradient factors Cm,LT incorporated into the
stiffness reduction functions only take into account the influence of the bending moment
shapes on the development of plasticity in this study. This approach is different to the usual
adoption of the moment gradient factors in traditional design, where they are generally used
for the determination of the most heavily loaded cross-section along the length of a member.
15
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT
My,Ed,2
My,Ed,1
My,Ed,3
My,Ed
My,Ed
NEd
Figure 8: Application of the stiffness reduction considering varying bending moments - tapering approach
16
1.0 1.0
1.0
GMNIA GMNIA
GMNIA
LBA-SR Cm,LT approach LBA-SR
LBA-SR Cm,LT approach
0.8 Cross-section
LBA-SR tapering strength
approach
0.8 LBA-SR tapering approach 0.8
Cross-section strength
Cross-section strength
y y
0.6 0.6
0.6
plpl
pl
N
N // N
/N
λ z = 0.4 λ z = 0.4 z
Ed Ed
z
Ed
N
HEA 240
N
0.4 0.4
0.4
IPE 240
λ z = 1.0 λ z = 1.0
0.2 0.2
0.2
λ z = 1.5 λ z = 1. 5
0 00
0 0.2 0.4 0.6 0.8 1.0 00 0.2
0.2 0.4
0.4 0.6
0.6 0.8
0.8 1.0
1.0
M /M M
My,Ed // M
My,pl
y,Ed y,pl y,Ed y,pl
Figure 9: Comparison of the flexural-torsional buckling strengths determined through the proposed stiffness
reduction method (LBA-SR) with those obtained through GMNIA for fork-end-supported beam-columns
subjected to uniform bending plus axial compression
of the proposed method for beam-columns with small, intermediate and large slendernesses.
In total, the strength predictions obtained through the proposed method were compared
against those obtained from GMNIA of the shell element models of 780 fork-end supported
beam-columns. The accuracy of the LBA-SR approach was assessed through the ratio ,
defined by eq. (18), where RLBA−SR and RGM N IA are the radial distances measured from
the origin to the interaction curves determined through LBA-SR and GMNIA respectively.
Values of greater than 1.0 indicate unconservative strength predictions. The comparison
of the results are presented in Table 5, where av and cov are the average and coefficient of
variation of values and min and max are the minimum and maximum values of values
respectively. Table 5 shows that both the LBA-SR Cm,LT approach and the LBA-SR taper-
ing approach provide accurate results, though the predictions are somewhat conservative for
cross-sections with h/b ≤ 1.2. For all considered sections, LBA-SR results in unconservative
errors of no greater than 4% (i.e. ≤ 1.04), indicating the reliability of the proposed stiffness
reduction method.
RLBA−SR
= (18)
RGM N IA
To assess the importance of the consideration of pre-buckling effects, LBA-SR was also
performed reducing the stiffness on the basis of the applied bending moments (first-order
bending moments) and axial compression. The comparison of the strengths obtained through
LBA-SR where pre-buckling effects are neglected against those of GMNIA are given in Table
17
Table 3: Range of European cross-section dimensions considered to assess the accuracy of the proposed
stiffness reduction method
Table 4: Number of beam-column cases examined to assess the accuracy of the proposed stiffness reduction
method
18
Table 5: Comparison of the results obtained through LBA-SR (i.e. the proposed stiffness reduction method)
and Eurocode 3 [1] Annexes A and B with those of GMNIA for fork-end beam-columns subjected to uniform
major axis bending plus axial compression
For the purpose of illustrating the accuracy of the stiffness reduction method in com-
parison to Eurocode 3 [1], the results obtained through the beam-column design methods
provided in Annexes A and B of Eurocode 3 [1] are also compared with those of GMNIA in
Table 5. The values were determined through the ratio of the radial distances measured
from the origin to the interaction curves determined from Annex A or Annex B and GMNIA
respectively. Note that the LTB formula given in the Clause 6.3.2.3. of Eurocode 3 [1] was
used in all the calculations in this paper. As seen from the table, the proposed stiffness
reduction method is more accurate than Annex B in almost all considered cases and is more
accurate then Annex A for beam-columns with aspect ratios larger than 1.2.
4.5. Application of the stiffness reduction method to beam-columns subjected to unequal end
moments or transverse loading
In this section, the accuracy of the proposed LBA-SR approach is assessed for fork-end-
supported beam-columns subjected to unequal end moments or transverse loading. The
strength predictions obtained through the LBA-SR Cm,LT approach and LBA-SR tapering
approach are compared against those determined through GMNIA and Eurocode 3 Annexes
A and B in Figs. 11 (a) and (b) for beam-columns with IPE 240 and HEA 240 cross-sections
and subjected to unequal end moments plus axial compression. Since all the beam-columns
analysed in this subsection have fork-end support conditions, the expressions provided in
19
0.6 NEd
GMNIA My,Ed
LBA-SR C approach
m,LT
LBA-SR tapering approach
LBA-SR pre-buckling effects neglected
0.4 GNA-SR in-plane analysis
Eurocode 3 Annex A 6 * 4.05 m
pl
/N
λ z = 1.00
N
y
λ LT = 0.81
0.2
z x
My,Ed
IPE 500
y
My,Ed
0
0 0.2 0.4 0.6 0.8 1.0 z NEd
M /M
y,Ed y,pl
Figure 10: Comparison of the flexural-torsional buckling strengths determined through the proposed stiffness
reduction method (LBA-SR) with those obtained through GMNIA for fork-end-supported beam-columns
subjected to uniform bending plus axial compression
Eurocode 3 [1] for the determination of the moment gradient factors are directly used in the
calculations to obtain strength predictions according to Eurocode 3 Annexes A and B. In
the LBA-SR Cm,LT approach, the quarter point formula given by eq. (17) is employed to
determine the moment gradient factors Cm,LT . As can be seen from the figure, the LBA-SR
Cm,LT and LBA-SR tapering approaches lead to very accurate strength predictions for the
case of single curvature bending given in Fig. 11 (a) and double curvature bending shown
in Fig. 11 (b). Though the Eurocode 3 [1] design methods are also in a good agreement
with the GMNIA results, the proposed stiffness reduction approaches provide more accurate
strength predictions.
Figs 11 (c) and (d) show comparisons of the strength predictions determined through the
LBA-SR Cm,LT and LBA-SR tapering approaches against those calculated through GMNIA
and Eurocode 3 [1] for beam-columns subjected to transverse loading and axial compression.
Note that transverse loading was applied at the shear centre of the beam-column cross-
sections in both cases. In the LBA-SR Cm,LT approach, the moment gradient factors Cm,LT
were calculated using the expressions provided by Kucukler et al. [27]. Since the beam-
columns are subjected to transverse loading, the increased imperfection factor αLT,F equal
to αLT,F = 1.4αLT was used for the determination of τLT,R given by eq. (13) in the LBA-
SR tapering approach. As can be seen from Fig. 11 (c) and (d), the LBA-SR Cm,LT
and LBA-SR tapering approaches provide results that are in a good agreement with those
obtained through GMNIA and are more accurate than Eurocode 3 [1], though the results are
rather conservative for beam-columns subjected to a point load at the mid-span and axial
compression. It should be noted that for the case of beam-columns subjected to uniformly
20
1.0 1.0
GMNIA GMNIA
LBA-SR Cm,LT approach LBA-SR Cm,LT approach
λ z = 0.4 λ z = 0.4
LBA-SR tapering approach LBA-SR tapering approach
0.8 Eurocode 3 Annex A
0.8 Eurocode 3 Annex A
Eurocode 3 Annex B Eurocode 3 Annex B
Cross-section strength Cross-section strength
0.6 0.6 λ z = 1.0
pl
pl
λ z = 1.0 y y
/N
/N
Ed
Ed
z z
N
N
My,Ed
0.2 My,Ed 0.2
0.5My,Ed 0.5My,Ed
0 0
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
M /M M /M
y,Ed y,pl y,Ed y,pl
y
pl
/N
/N
λ z = 1.0
Ed
z
Ed
z
N
My,Ed My,Ed
0 0
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
M /M M /M
y,Ed y,pl y,Ed y,pl
Figure 11: Comparison of the results obtained through Linear Buckling Analysis with stiffness reduction
(LBA-SR) with those obtained through GMNIA and Eurocode 3 (2005) for fork-end-supported beam-
columns subjected to different bending moment shapes and axial compression
distributed load, end moments and axial compression, shear forces become very significant in
some of the low slenderness (λz = 0.4) members, leading to shear failure, which limited the
maximum bending capacity. In the proposed LBA-SR approach, the shear capacity checks
were performed using the design rules given in Clause 6.2.6 of Eurocode 3 [1], which lead to
safe results for members failing due to shear as can be seen from Fig. 11 (c).
21
5. Application of the stiffness reduction method to irregular and multi-span
beam-columns
In this section, the accuracy of the proposed stiffness reduction method for the design
of irregular and multi-span beam-columns is investigated. In accordance with previous
sections, in-plane analysis GNA-SR is initially implemented as described in [16], and LBA-
SR is subsequently performed with reduced Young’s and shear moduli through τN,LT given
by eq. (14), considering section forces from GNA-SR. In all the considered cases, the results
obtained through the proposed approach are compared against those determined through
GMNIA using shell finite element models. Additionally, the results of the proposed stiffness
reduction method are also compared against those obtained from the beam-column design
methods of Eurocode 3 [1] so as to demonstrate the accuracy of the proposed approach
in comparison to traditional design. In traditional design, irregularities are accounted for
by calculating elastic buckling loads and moments and using these values in the associated
formulae such as those used for the determination of non-dimensional slendernesses. This
principle, which is referred to as the design by buckling analysis in AS 4100 [3] and implicitly
allowed in Eurocode 3 [1], is adopted herein so as establish a fair comparison between
traditional design and the proposed approach.
In the implementation of the design by buckling analysis (DBA) according to Eurocode
3 [1] for the assessment of the irregular and multi-span beam-columns performed herein,
Linear Buckling Analyses of the finite element models of the investigated members are
initially carried out considering only the corresponding component of loading that causes
pure compression or pure bending respectively. These Linear Buckling Analyses furnish
the elastic flexural buckling loads Ncr,y , Ncr,z , the elastic torsional buckling load Ncr,T and
the elastic critical moment Mcr of the investigated member, from which non-dimensional
slendernesses for flexural buckling λy , λz and that for lateral-torsional buckling λLT for the
most heavily stressed cross-section are determined. These slenderness values and elastic
buckling loads and moments are then used within the beam-column interaction equations
of Eurocode 3 [1], thereby assessing the utilisation rate of the most heavily stressed cross-
section. The imperfection factors used for the calculation of buckling reduction factors and
constants within the interaction equations that are functions of cross-section dimensions are
determined considering the cross-section properties of the most heavily stressed cross-section.
Following the recommendations of Goncalves and Camotim [32] and Boissonnade et al.
[33], exact equivalent moment factors are determined performing Geometrically Nonlinear
Analyses of the irregular and multi-span members investigated in this section and used
within the beam-column design equations of Eurocode 3 [1]. It should be noted that the
same principles followed in this paper were also adopted by Trahair et al. [34], Simoes
da Silva et al. [35] and Trahair [36] for the design of irregular and multi-span members
within steel frames according to the Eurocode 3 [1] and AS 4100 [3] provisions. Additional
information for the implementation of the DBA in conjunction with code provisions for
irregular and multi-span members can be found in these studies [34–36].
22
5.1. Stepped beam-column
In this subsection, the accuracy of the proposed stiffness reduction method for the design
of fork-end supported beam-columns strengthened with additional plates, which are referred
to as stepped beam-columns, is investigated. As shown in Fig. 12, a stepped beam-column
with an HEB 400 cross-section and subjected to uniform bending plus axial compression is
strengthened in the central half by attaching additional plates (t = 21.6 mm) to the flanges,
such that the second-moment area about the major axis is two times of that of the original
section. In the implementation of the stiffness reduction method, stiffness reduction was
applied to the original and strengthened portions considering corresponding cross-section
properties within the stiffness reduction functions. Note that warping moments and defor-
mations are fully transferred between the strengthened and original segments in the shell
finite element models of the stepped beam-columns. The flexural-torsional buckling loads
obtained through the proposed stiffness reduction method are compared against those ob-
tained from GMNIA in Fig. 12 for different slenderness values, where the non-dimensional
slendernesses λz,0 for the minor axis buckling are determined considering the original beam-
columns without the additional plates. Fig. 12 shows that both the LBA-SR Cm,LT and
LBA-SR tapering approaches provide very accurate results owing to the consideration of the
development of different rates of plasticity within the original and strengthened portions.
In addition to LBA-SR, the beam-column design methods of Eurocode 3 are also employed
considering the increased values of elastic buckling loads and moments. Fig. 12 illustrates
that the proposed stiffness reduction method brings about significant improvements in accu-
racy in comparison to traditional design, and that the strength predictions obtained through
Eurocode 3 [1] are often rather conservative.
L/4
Ed
L/4
N
0.4
L/2
λ z,0 = 1.5 Iy2=2.00Iy1, a a
0.2 HEB 400
Wpl,y2=1.87Wpl,y1
z
My,Ed
0 y
0 0.2 0.4 0.6 0.8 1.0
M /M a-a
y,Ed y,pl
Figure 12: Comparison of the flexural-torsional buckling strengths determined through the proposed stiffness
reduction method (LBA-SR) with those obtained through GMNIA for a stepped beam-column subjected to
uniform bending plus axial compression
1.0 NEd / 2
GMNIA
LBA-SR Cm,LT approach My,Ed
λ z,0 = 0.4
LBA-SR tapering approach
0.8 Eurocode 3 Annex A
Eurocode 3 Annex B
Cross-section strength
0.6 L/2
pl
NEd / 2
/N
λ z,0 = 1.0
Ed
N
0.4
L/2
a a
0.2 λ z,0 = 1.5
HEB 400
z
My,Ed
0
0 0.2 0.4 0.6 0.8 1.0 y
M /M a-a
y,Ed y,pl
Figure 13: Comparison of the flexural-torsional buckling strengths determined through the proposed stiffness
reduction method (LBA-SR) with those obtained through GMNIA for beam-column subjected to uniform
bending plus intermediate axial compression
24
to a fork-end supported beam-column with a discrete elastic lateral restraint at the mid-
height is investigated. The geometrical properties and loading conditions of the beam-
column with an HEB 400 cross-section and subjected to uniform major axis bending and
axial compression are illustrated in Fig. 14. It is seen that the elastic lateral restraint
is attached to the critical flange subjected to compressive stresses due to both bending
and axial load. To assess the accuracy of the design approaches for the consideration of the
influence of the lateral restraint, the stiffness of the elastic lateral restraint K was varied and
the strengths of beam-columns were determined though GMNIA, the LBA-SR Cm,LT and
LBA-SR tapering approaches and the beam-column design methods of Eurocode 3 Annexes
A and B. In the GMNIA simulations, two different shapes of geometrical imperfections were
considered: one-half sine wave and two-half sine waves, which correspond to the first and
second buckling modes. Moreover, non-proportional loading was applied in the GMNIA
simulations, where the axial loading was first applied up to 0.5Npl , and then the bending
moment was increased up to the collapse while the axial load was kept constant. The results
are compared in Fig. 14, where KL is the elastic threshold stiffness of the restraint leading
to the elastic flexural-torsional buckling of the beam-column in the second mode. Fig. 14
shows that up to a specific slenderness value, GMNIA with an imperfection in the form of
the first buckling mode shape leads to lower strength predictions than those obtained with
the imperfection in the form of the second buckling mode shape, while the latter provides
lower strength predictions after this value is exceeded. This specific stiffness can be referred
to as the inelastic threshold stiffness KL,inelastic leading to inelastic buckling of the beam-
column in the second-mode. With the development of plasticity in the beam-column, the
effectiveness of the lateral restraint increases as the ratio of the restraint stiffness to the
beam-column stiffness becomes larger [16]. Thus, the inelastic threshold stiffness KL,inelastic
leading to inelastic buckling in the second mode is considerably smaller than that for elastic
buckling KL . Fig. 14 shows that both the LBA-SR Cm,LT approach and LBA-SR tapering
approach are able to capture the described response and the increased effectiveness of the
elastic lateral restraint. Despite slight conservatism, both the LBA-SR Cm,LT and LBA-SR
tapering approaches also provide accurate strength predictions. A very important advantage
of the LBA-SR Cm,LT and LBA-SR tapering approaches is that they are able to capture the
transition between the first and second inelastic buckling modes directly without the need for
explicitly modelling the geometrical imperfections. Since the increased effectiveness of the
elastic restraint with the development of plasticity within the beam-column is not considered,
the beam-column design methods provided in Eurocode Annexes A and B cannot capture
the described response and lead to overly conservative results.
NEd=0.5Npl
0.5 My,Ed
ΚL,inelastic / ΚL
HEB 400
z
0.4
y
y,pl
/M
0.3 K
λ z,0 = 1.00
y,Ed
M
Figure 14: Comparison of the flexural-torsional buckling strengths determined through the proposed stiff-
ness reduction method (LBA-SR) with those obtained through GMNIA for a beam-column with an elastic
restraint at the mid-height
flexural-torsional buckling strengths of the beam-column were determined for different slen-
derness values λz,0 and different ratios of transverse loading to axial force through GMNIA,
the LBA-SR Cm,LT approach and LBA-SR tapering approach and the beam-column design
formulae given in Annexes A and B of Eurocode 3 [1]. In the GMNIA simulations, the
eigenmode corresponding to the lowest buckling load was adopted as an imperfection shape,
which was scaled by L/1000, i.e. 1/1000 of the length of the laterally unrestrained span.
The in-plane GNA-SR of the continuous beam-column was performed by reducing the stiff-
ness of each span separately through the stiffness reduction expressions and quarter point
moment gradient formula given in [16] on the basis of forces obtained through first-order
elastic analysis. After performing GNA-SR, the corresponding section forces were employed
to reduce the Young’s and shear moduli through τN,LT , and then LBA was performed in the
application of the LBA-SR Cm,LT and LBA-SR tapering approaches. In the LBA-SR Cm,LT
approach, the Cm,LT factors for each span were determined using the expressions developed
in [27]. Since the beam-column is subjected to transverse loading between lateral restraints,
the increased imperfection factor αLT,F = 1.4αLT was employed within τLT,R given by eq.
(13) in the application of the LBA-SR tapering approach. A comparison of the results is
displayed in Fig. 15, where the non-dimensional slendernesses for the flexural buckling λz,0
are determined assuming each laterally unrestrained segment as simply supported. Fig. 15
shows that the strength predictions obtained through both the LBA-SR Cm,LT approach and
26
the LBA-SR tapering approach are in a very good agreement with those determined through
GMNIA. For the studied continuous beam-column, the lowest segment of the member is sub-
jected to the largest forces and therefore experiences the greatest extent of plasticity. Since
the middle segment remains relatively elastic, the support afforded by this segment to the
critical lower segment effectively increases. Owing to the consideration of this behaviour
through the stiffness reduction functions, the LBA-SR Cm,LT and LBA-SR tapering ap-
proaches result in considerably more accurate strength predictions in comparison to those
obtained through Eurocode 3 [1] as can be seen in Fig. 15. It should be noted that Lim and
Lu [6] also investigated the response of continuous beam-columns, observing results similar
to those described herein. For the beam-columns with λz,0 = 0.4, the maximum bending
My,Ed capacity is limited by shear failure similar to the case illustrated in Fig. 11 (c). It is
seen from Fig. 15 that the use of the equations given in the Clause 6.2.6 of Eurocode 3 [1]
within the LBA-SR approaches to perform the shear capacity checks leads to safe results.
1.0
GMNIA NEd / 3
λ z,0 = 0.4 LBA-SR Cm,LT approach
LBA-SR tapering approach
0.8 Eurocode 3 Annex A
Eurocode 3 Annex B NEd / 3 L
Cross-section strength
0.6
pl
/N
w= NEd / 3
L2
N
0.4
w z
0.2 λ z,0 = 1.5 L
a a y
a-a
0
0 0.2 0.4 0.6 0.8 1.0 HEB 400
M /M
y,Ed y,pl
Figure 15: Comparison of the flexural-torsional buckling strengths determined through the proposed stiffness
reduction method (LBA-SR) with those obtained through GMNIA for a continuous beam-column
6. Conclusions
This study presented a stiffness reduction method for the flexural-torsional buckling
assessment of steel beam-columns. According to the proposed approach, the in-plane as-
sessment of a beam-column is initially carried out by performing Geometrically Nonlinear
Analysis with stiffness reduction (GNA-SR) where the section forces at the most heavily
loaded section are checked against the ultimate cross-section resistance as described in de-
tail in Kucukler et al. [16]. Provided the member does not fail in the in-plane analysis,
27
the out-of-plane assessment (flexural-torsional buckling assessment) is carried out by per-
forming Linear Buckling Analysis with stiffness reduction (LBA-SR). Stiffness reduction in
the out-of-plane assessment is applied on the basis of the section forces obtained from the
in-plane analysis (GNA-SR) through the stiffness reduction functions derived in this study.
For the consideration of the influence of moment gradient on the development of plasticity,
two alternative approaches may be employed: the LBA-SR Cm,LT approach and LBA-SR
tapering approach. The first is based on the incorporation of moment gradient factors into
the stiffness reduction functions, while the latter involves the division of a member into
portions along the length, with the reduction of stiffness for each portion based on the cor-
responding section forces. Shell finite element models of steel beam-columns were developed
and validated against experimental results from the literature. Geometrically and Materi-
ally Nonlinear Analysis with Imperfections (GMNIA) of these shell finite element models
furnished benchmark results used for the verification of the proposed stiffness reduction
method. The proposed method was verified against GMNIA results for 780 fork-end sup-
ported beam-columns subjected to uniform bending plus axial load covering 30 different
European IPE and HE shapes and different member slendernesses. The accuracy of the pro-
posed method for beam-columns subjected to unequal end moments and transverse loading
was also illustrated. In addition to regular and single span beam-columns, the proposed stiff-
ness reduction method was also assessed for irregular and multi-span beam-columns, where
the proposed method provided capacity predictions in very good agreement with those ob-
tained through GMNIA. It was also shown that since the influence of the development of
plasticity on the response of steel members is considered, the proposed stiffness reduction
method leads to considerably more accurate results in comparison to traditional design
based on the beam-column design methods of Eurocode 3 [1] for irregular and multi-span
beam-columns.
The proposed stiffness reduction method obviates the need of using member buckling
equations in design, considers compound buckling modes and assesses the out-of-plane in-
stability in a single step, so offering a realistic and practical way of designing steel members.
An important advantage of the proposed method is that it can be readily applied through
any conventional structural analysis software capable of providing elastic flexural-torsional
buckling loads for beam-columns. Future research will be directed towards the application
of the method to steel beam-columns with welded or monosymmetric cross-sections, those
susceptible to local buckling effects and to steel frames.
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