ARTICLE IN PRESS

International Journal of Mechanical Sciences 50 (2008) 1280– 1291

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International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

An alternative approach to the analytical determination of the moment

capacity of a thin-walled channel steel section beam
Harkali Setiyono 
The Technology Center for Structural Strength, Agency for the Assessment and Application of Technology (B2TKS-BPPT), Kompleks PUSPIPTEK, Serpong, Tangerang 15314, Indonesia

a r t i c l e in f o a b s t r a c t

Article history: An analytical method of combined plastic mechanism and elastic approaches has been developed to
Received 24 October 2007 predict the moment capacity of a thin-walled channel steel section beam. The plastic mechanism
Received in revised form approach is performed by analyzing an idealized model of bending-plastic-hinge-collapse mechanism
7 April 2008
of the beam bent about its minor neutral axis. This approach adopts a concept of the equilibrium
Accepted 15 May 2008
between external energy and the one dissipating in the hinge mechanism. Another analytical approach
has been done by analyzing the investigated beam according to an elastic-bending theory. In the elastic
Keywords: analysis, the application of an effective width concept has also been considered to account for the effect
Channel section of local buckling on the bending element of the beam. These both analytical approaches are then used in
Plastic-hinge mechanism
the method of cut-off strength to estimate a theoretical moment capacity of the beam bent about its
Moment capacity
minor neutral axis. An attempt has also been carried out to correlate the estimated moment capacity
Local buckling
Bending and effective width obtained to another one about a major neutral axis. In order to assess the accuracy of the analytical
method developed, its predicted results are verified using the data obtained from experiments and a
design code specially used for cold-formed steel structural members. The data of moment capacity
about the minor neutral axis is compared to the one measured in a number of flexural-bending tests to
failure on the similar channel steel section beams. Meanwhile, the data of moment capacity about the
major neutral axis is compared to the one calculated using the design code. A statistical analysis of
verified data populations indicates that the mean value of deviated data from experimental one is 1.025
with the standard deviation of 0.087 and from the design code is 1.004 with the standard deviation of
0.111. These statistical measures clearly mean that the analytical model presented herein, on average,
tends to estimate conservatively the moment capacity of the investigated beam about its both
unsymmetrical and symmetrical–neutral axes by less than 5% and this is certainly still within
acceptable limits of 720%.
& 2008 Elsevier Ltd. All rights reserved.

1. Introduction In the elastic approach, the investigated beam is analyzed

according to a theory of elastic-bending beam and section proper-
An alternative-analytical model of combined plastic mechan- ties of the beam are determined by adopting an effective width
ism and elastic approaches has been developed to predict the concept. The application of the effective width concept in the
moment capacity of a thin-walled channel steel section beam. The analysis is intended to account for the effect of local buckling in
plastic mechanism approach is performed by analyzing a bending the bending element. Because the existence of local buckling, the
failure mechanism model of the investigated beam as shown in affected element is no longer fully effective in carrying applied
Fig. 1 and the analysis is based on a concept of energy equilibrium. loads. Accordingly, it is necessary to consider the dimension of the
In this concept, the external energy affected by applied-bending element that is still effective in carrying loads in the elastic analysis.
moments is equated to the sum of energy absorbed by the Both analytical approaches can generate two different-theore-
bending failure mechanism model. The bending failure mechan- tical load–deflection curves, i.e. an unloading load–deflection
ism model can be recognized from a very careful observation on a curve and another inclining load–deflection one. The former is
number of thin-walled channel steel section beams that have obtainable from the plastic mechanism approach and the latter is
been tested under flexural-bending moments to failure. obtainable from the elastic one. These theoretical load–deflection
curves are then used in the method of cut-off strength as shown in
Fig. 2 to estimate a moment capacity of the investigated beam.
The application of the method of cut-off strength in this
 Tel.: +62 217560562x1043; fax: +62 217560903. research program is mainly aimed at estimating the moment
E-mail addresses: harkali@luk.or.id, harkali_setiyono@yahoo.co.id. capacity of the investigated beam, which is bent about its minor

0020-7403/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmecsci.2008.05.004
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H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1281

Nomenclature Mcx moment capacity of the investigated channel beam

about its minor neutral axis
bf full width of the flange element Mcy moment capacity of the investigated channel beam
bw overall height of the web element about its major neutral axis
E modulus of elasticity Mel elastic moment carrying capacity
Fk correlating factor Mp moment capacity of plastic hinge, which is based on
Fy design yield strength fully plastic moment
Ixeff second moment of the effective cross-section about Mpl plastic moment carrying capacity
its minor neutral axis Mnb nominal flexural strength
Iyeff second moment of the effective cross-section about Mnt lateral–torsional buckling strength
its major neutral axis rc inside corner radius between flange and web ele-
Iy–y second moment of the full cross-section about its ments
major neutral axis t wall thickness of the investigated channel beam
Kf, Kw buckling constant of the unstiffened flange or stif- y rotation angle of plastic hinge during deformation
fened web elements Z mid-span deflection of the investigated channel beam
L or l overall length of the investigated channel beam or f mechanism rotation angle
plastic hinge sy yield strength of the basic material
M applied bending moment

Flange elements

Web element

6 8 79 7
10
10
8 5
2 6 2
1 3 4 9 3
5 1 4

1 – 10 : plastic hinges

Plastic hinge collapse mechanisms

Fig. 1. Bending-plastic failure mechanisms of the investigated beam [1]: (a) side view and (b) top view.

neutral axis with free edges of the flange elements in compres-

Load

sion. In Fig. 2, the value of load at the point of cut-off strength is

assumed the maximum load that can be carried by the beam prior Elastic curve
to failure. This maximum load obtained in this research program
is certainly equivalent to the value of theoretical moment
capacity.
Fig. 3 illustrates the terminology of main dimensions in the Cut-off strength
cross-section of the investigated beam that is used throughout
Pe : Elastic limit load
this research program. As mentioned above, the effect of flexural Pf Pm
Pf : Ultimate limit load
bending causes properties of the cross-section shown in Fig. 3 to Pe
Pm : Plastic limit load
be determined with respect to the minor neutral axis and the
Actual behaviour
moment capacity is investigated analytically and experimentally.
Plastic mechanism curve
In order to study further, the application of the analytical model
developed, the predicted moment capacity obtained is attempted
to be correlated to another moment capacity of the beam bent (0, 0) Deflection
about its major neutral axis.
Fig. 2. Method of cut-off strength [2–5].
The accuracy of the analytical model developed is assessed by
verifying its predicted results. The verification is carried out by
comparing the analytical predictions to the data of moment
capacity measured in a number of flexural-bending tests on the predicted data to the one of tests and the design code are limited
similar beams as well as the one calculated using the design code within acceptable scatter tolerances of 720% and statistically
of cold-formed steel structural members. Scattered deviations of evaluated. Beyond the accuracy assessment of the predicted
 ARTICLE IN PRESS

1282 H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291

Major or symmetrical-neutral axis Minor or unsymmetrical-neutral axis

Center M
t 7
M 10
Wf 6 8 9
Wf bf
2 5
rc φ α1 3 4
Ww = bw – t Center
Wf = bf – 0.5t Ww η β
2a
Ww
L
bw

Fig. 3. Main dimensions of the cross-section.

Fig. 4. Schematic drawing of bending-plastic failure mechanisms.

moment capacity obtained, the analytical model developed is also

used to assess the effect of geometrical parameter such as bending Compressive stress region
elements of the beam on the predicted moment capacity. The
accuracy of this assessment is measured by verifying the
M
obtainable results using the ones of the flexural-bending tests as yc
well as the design code. M
Wf 2 yc
1
2. Analytical approach

In the analytical approach, two different methods of analysis α 3

are employed to establish theoretical curves of load–deflection Tensile stress region φ β yt
behavior of the investigated beam, which is subjected to flexural
bending about its minor neutral axis. These curves represent the a
behavior of moment carrying capacity and deflection relationship
of the beam obtained from plastic mechanism and elastic Fig. 5. Analytical determination of the plastic hinge length.
analyses. The plastic behavior of the moment–deflection relation-
ship is in the form of an unloading curve whereas the elastic one is
in the form of an inclining curve. With reference to the method of
cut-off strength shown in Fig. 2, the moment capacity with where Eext is the external energy produced by the action of
respect to the minor neutral axis can be directly estimated from applied-bending moment; Mp is the moment capacity of each
the value of moment carrying capacity at an intersection of the plastic hinge; and y is the rotation angle of each plastic hinge
plastic and elastic behavior curves. during deformation.
The plastic mechanism analysis is applied to the model of In Fig. 4, each plastic hinge is in a symmetrical condition so
bending-plastic failure mechanisms displayed in Fig. 1. This that the energy equilibrium Eq. (1) can be rewritten as follows:
analysis will result in an equation of plastic moment expressed
Mf ¼ 2fðMp Þ1 y1 þ ðM p Þ2 y2 þ ðM p Þ3 y3 g (2)
in terms of deflection at the mid-span of the beam. In the elastic
analysis, the beam is treated as an elastic-bending one and the As axial forces of the bending stress carried by the flange
effect of local buckling on the bending elements is taken into elements is not uniformly distributed, the moment capacity
account by adopting an effective width concept in determining its of each plastic hinge (Mp) is therefore determined according
section properties. Another expression of elastic moment–deflec- to the fully plastic moment alone with the absence of the
tion relationship is obtained and this is then iterated along with axial force effect. The following is a formula of the fully plastic
the plastic one using a computer program specially written for moment used to represent the moment capacity at each plastic
this purpose. The iteration process is terminated as the plastic hinge [3,6]:
moment has been converged to the elastic one at a point and the
value of moment capacity can be directly estimated from this sy t 2i li
Mp ¼ (3)
point. The analytical approach is subsequently performed to 4
correlate the moment capacity about the minor neutral axis to In Eq. (3), ti and li are the thickness and length of each plastic
another one about the major neutral axis. hinge respectively, while sy is the material yield strength. The
thickness ti is equal to the wall thickness of the investigated beam
2.1. Plastic mechanism analysis and the length li is obtained from the analytical determination
indicated in Fig. 5.
It is clearly obtainable from Fig. 5 that the length of the plastic
Fig. 4 is a sketch representing the bending-plastic failure
hinges 1–3 can be calculated using Eq. (4). Referring to Fig. 4, the
mechanism model shown in Fig. 1 and this is used to develop the
length of the plastic hinge 1 is actually equal to one of the plastic
plastic mechanism analysis. This analysis is based on a concept of
hinges 5, 6 and 10. Meanwhile, the plastic hinges 2 and 7 are
energy equilibrium where the external energy produced by the
similar in length and this similarity is also for the plastic hinge 3
action of applied-bending moment is equated to the sum of energy
and the rest of the other three ones:
dissipating in the plastic hinge mechanism. A basic formulation of
this energy equilibrium can be written as follows [5]: Wf W ðtan a  tan bÞ
l1 ¼ l5 ¼ l6 ¼ l10 ¼ ; l2 ¼ l7 ¼ f
X
i¼n sin a tan a
Eext ¼ ðM p Þi yi (1) Wf
l3 ¼ l4 ¼ l8 ¼ l9 ¼ (4)
i¼1 tan a cos b
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H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1283

Having determined the length of each plastic hinge, the analytical 2.2. Elastic analysis
determination is subsequently employed to determine the rota-
tion angle of each plastic hinge during deformation. The formula In the elastic analysis, the investigated beam is treated as an
of each angle is as follows: elastic-bending beam theory and analyzed according to a non-
rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi linear material stress–strain relationship. The expression of this
2 sin a f cos a 2f tan a
y1 ¼ ; y2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , relationship is as follows [5,7]:
sin ð2a þ fÞ tan a f1 þ cos ð2a þ fÞg
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1    
2 2 s 3s s n1 log 17
x2  x1 ¼ þ ; n¼1þ  7  (15)
y3 ¼ tan1 @ A (5) E 7E s0:7 log ss0:85
0:7
x1
The symbols of s, e and E represent the stress, strain and elastic
The factors of x1 and x2 in the equation of y3 is computed from:
modulus of the basic material. Meanwhile, s0.7 and s0.85 are,
x1 ¼ ðB1 FÞ2 þ ðC 1 FÞ2  ðB1 C 1 Þ2 ; x2 ¼ 2ðB1 FÞðC 1 FÞ (6) respectively, the stresses corresponding to 70% and 85% of the
elastic modulus E. In the above equation of e, the first right hand
W f tan ða  bÞ W f tan b side term represents the linear-elastic behavior of the basic
B1 F ¼ ; C1F ¼ ,
tan a cos b tan a cos b material and the second one represents the non-linear-elastic one.
Wf An elastic moment carrying capacity of the beam shown in
AB1 ¼ (7) Fig. 6 is obtainable from deriving Eq. (15) and expressed in terms
tan a cos b cos ða  bÞ
of the mid-span elastic deflection (Z). Fig. 6 clearly indicates that
Wf the applied-bending moment M causes the bending element of
AC 1 ¼ ,
tan a cos2 b the beam to be more affected by compressive stresses than tensile
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ones. It is therefore the derivation of Eq. (15) to be performed
B1 C 1 ¼ ðAB1 Þ2 þ ðAC 1 Þ2  2ðAB1 ÞðAC 1 Þ cos a (8)
according to the stress and strain in the compression region and
Substitution of Eqs. (3)–(8) into Eq. (2) and further expanding it, their values are:
the plastic moment carrying capacity of the beam (Mp) can be 8yc ZE Myc
directly expressed in terms of the mechanism angle (f): sc ¼ ; c ¼ (16)
L2 EI
M p ¼ C 1 ðC 2 þ C 3 þ C 4 Þ (9) Eq. (16) is then substituted into Eq. (15) and analyzing it to
where formulate the following elastic moment carrying capacity:
 
sy t 2 W f 8EIZ 24EIZ 8Eyc Z n1
C1 ¼ , M¼ 2
þ (17)
2 tan a sin a cos bðfÞ L 7L2 L2 s0:7
rffiffiffiffiffiffiffiffiffiffiffi
2 tan a cos b sin a f Eq. (17) consists of two different elastic behaviors, i.e. linear and
C2 ¼ ; fX0:001 (10)
sin ð2a þ fÞ tan a non-linear elastic behaviors. The elastic moment carrying capacity
(Me) is determined according to these linear and non-linear elastic
C 3 ¼ sin aðy3 Þ, behaviors. In order to account for the effect of local buckling in
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2f tan a bending elements of the beam, section properties used in the
C 4 ¼ sin a cos bðtan a  tan bÞ cos a elastic analysis are determined using the effective cross-section.
f1 þ cos ð2a þ fÞg
Thus, the linear-elastic moment carrying capacity (Me) is
fX0:001 (11)
calculated using:
With reference to Fig. 4, the mid-span deflection of the beam (Z)
8EIxeff Z
can be expressed in terms of the mechanism angle (f) using the Me ¼ (18)
small deflection theory. Thus, the plastic mid-span deflection is L2
In the meantime, the non-linear elastic moment carrying capacity
ðL  2aÞf
Z¼ (12) (Mne) is
2
 
The angles of a and b can be experimentally recognized and, on 24EIxeff Z 8Eyceff ðZne  Z0:7 Þ n1
Mne ¼ M0:7 þ (19)
average, they are close to 751 and 601, respectively. In the meantime, 7L2 L2 s0:7
the value of (a) approximately approaches to (Wf/tan a) and a
where Me and Mne represent the elastic moment carrying capacity
reduction factor Rf should be introduced to reduce the plastic
(Mel) and, Ixeff is the second moment of area about the minor
moment carrying capacity (Mpl). Its formulation can therefore be
neutral axis of the effective cross-section; yceff is the position of
rewritten as shown in Eq. (13). The reason of introducing Rf is to
compressed outer fiber from the minor neutral axis of the
produce an analytical moment capacity, which is close to actual one
effective cross-section; M0.7. Is the elastic moment carrying
measured in experiments. In this research program, Rf is determined
based on the square root of the ratio between section modulus of
the full cross-section in the tensile region and the compressive one.
A further analysis of this square root of the section modulus ratio Compressive stress distribution
M M
will provide an expression used to determine Rf in Eq. (14): yc σc
M pl ¼ Rf C 1 ðC 2 þ C 3 þ C 4 Þ (13)
yt
rffiffiffiffiffi 2
η
yt ðW f  rÞðW f þ rÞ þ 4r
Rf ¼ ; yt ¼ ; yc ¼ W f  yt (14)
yc 2ðW f  rÞ þ pr þ W w
Eqs. (13) and (14) are subsequently used in the iteration process to σt
L Tensile stress distribution
establish an unloading theoretical bending–deflection relation-
ship curve. Fig. 6. Elastic-bending beam theory.
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1284 H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291

Full-unstiffened flange Effective-unstiffened flange

(bending) elements (bending) elements

Center
Center

y Wf1
y Weff

yc L
Wf x x yceff
yt r L x yteff r x

r = rc + 05t 1
y Ww y Ww x-x : minor neutral axis
y-y : major neutral axis

Stiffened web (tensile)

element

Fig. 7. Full and effective section dimensions: (a) full section and (b) effective section.

capacity corresponding to s0.7; Zne is the mid-span deflection at curve of an inclining-elastic moment–deflection relationship will
the non-linear elastic region; and Z0.7 is the mid-span deflection be established. In order to implement the method of cut-off
corresponding to M0.7. strength shown in Fig. 2, Eqs. (13), (14), (18) and (19) are iterated
The effective section properties (Ixeff and yceff) are determined to estimate the moment capacity of the investigated beam bent
from the effective section of the beam in Fig. 7(b). An effective about its minor neutral axis (x–x). The iteration is carried out by
width concept used to account for the effect of local buckling is initially setting the value of f ¼ 0.001 and incrementally increas-
only applied to the compressive region of the bending elements. ing it until the value of plastic moment carrying capacity (Mpl)
Meanwhile, the other tensile ones are assumed still fully effective and the elastic one (Mel) converge at a point. The value of moment
in carrying loads. carrying capacity at this converging point is assumed the
In Fig. 7(b), W 1f and W 1w are equal to (Wfr) and (Ww2r), theoretical moment capacity about the minor neutral axis.
whereas Weff is the effective width of flange elements and it is An attempt has also been employed to correlate the theoretical
determined according to following procedure [8]. If the slender- moment capacity about the minor neutral axis to another
ness factor of flange elements (lf) is moment capacity of the beam about the major one. This corre-
lation is determined according to a criterion of conventional–
lf p0:673 ) W eff ¼ W 1f ; lf 40:673 ) W eff ¼ rW 1f (20) structural design. In this criterion, the applied load should be
lf and r are calculated using: designed in such a way that it is not permitted to cause plastic
sffiffiffiffiffiffi deformation within the structure. In order to be able to correlate
f 1  ð0:22=lf Þ these two moment capacities, the following reasons are taken into
lf ¼ ; r¼ (21)
F cr lf account in the analysis:

where f is a stress carried by compressed parts of the bending

 The moment capacity is actually the maximum moment that
flange elements and it can be seen in Fig. 6 that the maximum
can be carried by the structure prior to failure.
compressive stress is greater than the tensile one (sc4st).
 The structure is considered to fail by plastic deformation as the
Accordingly, the initial yielding will occur in the compressive
applied-bending moment has approached to a fully plastic
region so that for the purpose of analysis, the value of f can be
moment (Mp).
equated to the yield strength (sy).
Fcr is a critical buckling stress of the unstiffened flange
elements and this stress is determined as follows: Referring to these above reasons, the correlation of these two
moment capacities is formulated as follows:
!2
p2 E t  
F cr ¼ 0:43 (22) ðM cx Þth ðM p Þy ðM yield Þy 0:05
12ð1  m2 Þ W 1f ðM cy Þth ¼ F k ; Fk ¼ (25)
ðM p Þx ðMyield Þx
where E and m are an elastic modulus and Poisson’s ratio of the where (Mcy)th is the theoretical moment capacity about the major
basic material respectively, and the value of Weff obtained is then neutral axis; (Mcx)th is the theoretical moment capacity about the
used to calculate the other section properties such as yteff, yceff and minor neutral axis; (Mp)x is the fully plastic moment about
Ixeff. These properties are expressed in Eqs. (23) and (24): the minor neutral axis; (Mp)y is the fully plastic moment about
W eff ðW eff þ 2rÞ þ r 2 ðp  2Þ the major neutral axis; and Fk is the correlating factor.
yteff ¼ , The introduction of the correlating factor (Fk) in Eq. (25) is
2W eff þ pr þ W 1w
intended to provide the value of (Mcy)th, which is close to the one
yceff ¼ W eff þ r  yteff ; r ¼ r c þ 0:5t (23)
of the design code. This factor is expressed in terms of the ratio
2 h  
2 i 3 between first yield moments of the beam bent about its major and
2pW eff W 2eff þ 3 W eff  2 yteff þ r minor neutral axes, i.e. (Myield)y and (Myield)x. The values of
6 h i7
t 66 þ 6r r 2 p2  8 þ 2p 2r þ pW  y 
2 7
7 (Myield)x, (Myield)y, (Mp)x and (Mp)y are determined by considering
Ixeff ¼ (24)
12p 6
4
eff ceff 7
5 the effect of local buckling in the bending elements of the beam
 
þ p W 1w t 2 þ 12y2teff and their expressions can be seen in Eqs. (26) and (27):

Substituting Eqs. (23) and (24) into Eqs. (18) and (19) and further sy Ixeff sy Iyeff
ðM yield Þx ¼ ; ðM yield Þy ¼ (26)
analyzing them through an iteration process, another theoretical yceff ycy þ 0:5t
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H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1285

2 3
y2ceff þ ðyteff  rÞ2 þ rf2r þ pðyteff  rÞgþ where ff and fw are stresses acting on the part of the flange and
ðMp Þx ¼ sy t 4 5 web elements, which initially undergoes yielding. It can be seen in
ðW w  2rÞyteff
the bending stress distribution of Fig. 8(b) that the maximum
r ¼ r c þ 0:5t compressive stress (sc)max is greater than the tensile one (st)max.
2 r fpycy rðp2Þg
3 Therefore, initial yielding will certainly be taken place on the
ðW eff Þy ycy þ 2 þ
6 n o 7 compressive parts of the flange and web elements. For the
6 7
6W 2 7 purpose of analysis, the value of ff and fw is equated to the yield
6 efw1 ycy  ð0:5W efw1 þ rÞ þ 0:5W efw2 7
6
ðMp Þy ¼ sy t 6 7 (27) strength of the basic material (sy) and the critical buckling stress
7
6 2 r fpyty rðp2Þg 7 of flange (Fcr)f as well as web (Fcr)w is
6 þ0:5ðyty  rÞ þ 2 7
4 5
þyty ðW f  rÞ !2
p2 E t
ðF cr Þf ¼ 0:43 ,
Effective dimensions such as (Weff)y, Wefw1 and Wefw2 are clearly 12ð1  m2 Þ W 1f
presented and discussed in the next section. !2
p2 E t
ðF cr Þw ¼ kw (30)
12ð1  m2 Þ W 1w
3. Application of the design code
where E and m are the elastic modulus and Poisson’s ratio
In this research program, the application of the design code is respectively, whereas kw is a buckling coefficient of the stiffened
mainly aimed at determining the moment capacity of the web element. The calculation of this coefficient is as follows:
investigated beam when it is bent about its major neutral yt1
axis (Mcy). The design code used is the one that is specially kw ¼ 4 þ 2ð1 þ cÞ3 þ 2ð1 þ cÞ; c¼ (31)
yc1
recommended for design analyses of cold-formed steel structural
members [8]. In case of the flexural-bending beam investigated
2ðW 1w þ 2rÞðW eff Þy þ ð2r þ pW 1w þ prÞr þ ðW 1w þ 2rÞW 1w þ 2r 2
here and according to the design code that the moment capacity yt1 ¼ ,
(Mcy) should be determined from the smallest value of: 2ððW eff Þy þ pr þ W 1w þ W 1f Þ
yc1 ¼ W w  yt1 (32)
 Nominal flexural strength (Mnb);
 Lateral–torsional buckling strength (Mnt). r ¼ r c þ 0:5t; W 1f ¼ bf  ðr c þ tÞ; W 1w ¼ bw  2ðr c þ tÞ (33)

The nominal flexural strength (Mnb) is determined by taking The effective width of the top flange element (Weff)y is
into account the effective cross-section shown in Fig. 8 and calculated from Eq. (34) and the value of rf is as indicated in
formulated as follows: Eq. (35). In case of the bending web element, its effective width is
determined by the procedure as specified in Eqs. (36)–(38):
M nb ¼ Se F y (28)
Se is an elastic section modulus of the effective cross-section and lf p0:673 ) ðW eff Þy ¼ W 1f
Fy is the design yield strength that is equivalent to the yield lf 40:673 ) ðW eff Þy ¼ rf W 1f (34)
strength of the basic material.
In Fig. 8, (Weff)y, Wefw1 and Wefw2 are the effective width of the  
compressed-unstiffened flange and the bending-stiffened web. 1  0:22
lf
rf ¼ (35)
These effective dimensions are determined according to the lf
slenderness factor of the flange (lf) and web (lw) elements, where
the factor can be obtained from: lw p0:673 ) W efw ¼ W 1w ; lw 40:673 ) W efw ¼ rw W 1w
sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi  
ff fw 1  0:22
lw
lf ¼ ; lw ¼ (29) rw ¼ (36)
ðF cr Þf ðF cr Þw lw

bf (Weff)y
Wf
(σc)max (σc)max

rc x
Wefw1 rc x
0.5Ww
t ycy
y Center y
bw Ww bw Ww1 Wefw2 Ww
Center
y y
0.5Ww t
yty
rc x rc x

(σt)max (σt)max
y0 Wf1

Fig. 8. Full and effective cross-sections: (a) full cross-section and bending stress distribution and (b) effective cross-section and bending stress distribution.
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1286 H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291

W efw F e X2:8sy ) F c ¼ sy ,
W efw1 ¼ , 
3þc 10 10sy
8 2:78sy 4F e 40:56sy ) F c ¼ sy 1  ,
bw < c40:236 ) W efw2 ¼ W efw 9 36F e
2
p4 ) (37) F e p0:56sy ) F c ¼ F e (43)
bf : cp0:236 ) W efw2 ¼ W efw  W efw1

cb p2 EðW w  tÞIxc
Fe ¼ (44)
bw W efw Sf ðK x lx Þ2
44 ) W efw2 ¼ (38)
bf ð1 þ cÞ  W efw1 In Eq. (44), the value of cb may be equated to unity for the reason of a
conservative approach and the other factors are as follows:
The subsequent analysis is to determine section properties of the E is the elastic modulus; Ixc is the second moment of compression
effective cross-section shown in Fig. 8(b). It can be seen in Figure region of full cross-section shown in Fig. 8(a) about the minor neutral
that the neutral axis y–y of the effective cross-section is no longer axis x–x and it can be calculated using Eq. (45); Sf is the compressive-
a symmetrical one and its position is as follows: elastic section modulus of full cross-section and it is formulated in
Eq. (47); Kx is the effective length factor and if ends of the beam can
2yt1 ððW eff Þy þ pr þ W 1w þ W 1f Þ  f2W 1w þ 2r  yc1 þ ðW efw1 þ W efw2 Þ  2W efw1 g
yty ¼ freely rotate to the direction of its minor, major and longitudinal axes,
2fðW eff Þy þ pr þ W 1w þ W 1f  yc1 þ ðW efw1 þ W efw2 Þg
the value of Kx ¼ 1.10 [9]; and lx is the overall length of the beam
yc1  ðW efw1 þ W efw2 Þ
 , without bracing and it is bent about its minor neutral axis x–x.
2fðW eff Þy þ pr þ W 1w þ W 1f  yc1 þ ðW efw1 þ W efw2 Þg 
2 n o 3
(39) 1 1 2 1 2 2 2
6 2pW f ðW f Þ þ 3ðW f þ 2r  2y0 Þ þ 6r r ðp  8Þ 7
ycy ¼ W w  yty
6 7
Ixc ¼ t 6 n o2  7
yt1 and yc1 in the above equation are the same as the ones 4 5
þ2 py0  rðp  2Þ þ pW 1w ðt 2 þ 12y20 Þ
computed using Eq. (32). Meanwhile, the elastic section modulus
(Se) in Eq. (28) is calculated about the outer fiber of the (45)
compressive region so that its value is
where y0 is the distance of the minor neutral axis x–x to the centerline
Iyeff of the web element (see Fig. 8) and it can be computed using:
Se ¼ (40)
ycy þ 0:5t W 1f ðW 1f þ 2rÞ þ r 2 ðp  2Þ
y0 ¼ (46)
2W 1f þ pr þ W 1w
Iyeff is the second moment of the effective cross-section about the
neutral axis y–y and it can be obtained from: Iyy
Sf ¼
0:5ðW w þ tÞ
2 n o  3
ðW eff Þy tðt 2 þ 12y2cy Þ 1 2 1 2 2 2
Iyeff ¼ 6 2pW f t þ 3ðW w þ 2rÞ þ 3r 2r ðp  8Þ 7
12 t 66
7
7
Iyy ¼ n o  (47)
trbr 2 ðp2  8Þ þ 2fpycy  rðp  2Þg2 c 6
12p 4 1
2
1 3
7
5
þ þ pðW þ 2rÞ  2rðp  2Þ
w þ pðW Þ w
4p
tW efw1 fW 2efw1 þ 3ð2ycy  W efw1  2rÞ2 g tW 3efw2
þ þ Hence, the moment capacity of the investigated beam bent about
12 3
3
its major neutral axis y–y (Mcy) analytically obtained from the
tðyty  rÞ tr½r ðp  8Þ þ 2fpyty  rðp  2Þg2 
2 2
design code is as follows:
þ þ
3 4p (
1 2 Mnb 4M nt ) M cy ¼ M nt
tW f ðt þ 12y2ty Þ if (48)
þ (41) Mnb oM nt ) M cy ¼ M nb
12

Substitution of Eqs. (40) and (41) into Eq. (28) will provide the
value of nominal flexural strength (Mnb) of the investigated beam
Table 1
subjected to flexural-bending moments about its major neutral
Mechanical properties of the basic material
axis y–y.
The lateral–torsional buckling strength (Mnt) is determined Specimen group sy (MPa) s0.7 (MPa) s0.85 (MPa) sUTS (MPa) E (  103 MPa)
using Eq. (42) where Sc is equivalent to Se formulated in Eq. (40).
t ¼ 1.60 mm 153.00 115.33 93.67 291.00 191.67
Fc can be calculated according to the procedure as indicated in
t ¼ 2.30 mm 175.00 162.66 145.33 314.00 194.81
Eq. (43). In this equation, Fe is an elastic-critical lateral–torsional t ¼ 3.20 mm 266.00 252.18 240.20 371.00 198.89
buckling stress and obtainable from Eq. (44):
sy, yield strength; s0.85, stress corresponding to 85% E; E, modulus of elasticity; s0.7,
M nt ¼ Sc F c ; Sc ¼ Se (42) stress corresponding to 70% E; sUTS, ultimate tensile strength.

r = 20.00
A t

20.00 40.00

22.36 A 22.36
40.00 120.00 40.00 Cross section A-A
244.72

Fig. 9. Design of tensile specimen [5,10,11].

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H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1287

bw Nominal dimensions:
bw = 100 and 150 mm
Ww bf = 40 and 50 mm
t = 1.60, 2.30 and
3.20 mm
rc rc = 2.25 mm
bf Wf t L = 600 mm
90°° 90°
L Ww = b w – t
Wf = bf – 0.5t

Fig. 10. Design of flexural-bending tested specimen.

BMD : Bending Moment Diagram

BMD Fe : test load generated by RME 100
Fe
LVDT 1 y LVDT 2
Fe / 2 LVDTs 1 & 2 A Fe / 2

x Fe / 2 Fe / 2 t rc x bf
A bw y

110 370 110 Cross section A-A

L = 600

Fig. 11. Schematic illustration of flexural-bending tests.

4. Experimental approach Testing machine RME 100 Schenck Trebel LVDT

The experimental approach was carried out in two steps, which

consisted of tensile tests of the basic material and flexural-bending

End supports of test load reaction

tests on 15 specimens of thin-walled channel steel section beam. In
the tensile tests, tensile specimens tested are made of a carbon steel
Pressure devices

material of Standard JIS G 3141-SPPC and designed according to

Standard JIS Z 2201 no. 13A. Fig. 9 shows the design of tensile
specimens, which are classified into three groups of thickness, i.e. t
is equal to 1.60, 2.30 and 3.20 mm. Each group consisted of three
identical specimens and they were tested to failure at ambient
temperatures on a static testing machine RME 100 Schenck Trebel.
Mechanical properties obtained from the tensile tests on each group
of specimens are displayed in Table 1 and these properties are used
as material data in the analytical approach.
The specimens tested under flexural-bending moments are Test specimen
manufactured from the carbon steel sheet of Standard JIS G 3141-
Fig. 12. Configuration of flexural-bending tests.
SPPC through cold-forming processes. Design of these specimens
is depicted in Fig. 10 and manufactured in various nominal
dimensions. In the flexural-bending tests, each specimen was achieved. With reference to this maximum test load, the
tested in the form of a four-point loading beam to failure. Fig. 11 maximum moment carried or the actual moment capacity of
shows the schematic diagram of the test arrangement and the each tested specimen is determined and then used to verify the
actual flexural-bending test arrangement, which was conducted theoretical moment capacity of the analytical model. Beyond the
on the static testing machine RME 100 Schenck Trebel, can be seen measurement of actual moment capacity, a failure mechanism
in Fig. 12. The test load was applied to each specimen in such a mode of each tested specimen was also carefully observed. It has
way that the area of the specimen affected by maximum bending been repeatedly found from the tests that the failure mechanism
moments was kept at a constant length of 370 mm. of the specimen subjected to this type of flexural-bending
During the tests, a relationship of test load and mid-span moments is the same as that shown in Fig. 1.
vertical deflection of each specimen was always monitored and
recorded by using data logger and X–Y plotter. The test load and
mid-span vertical deflection can be directly measured from the 5. Verification of the analytical model
testing machine and LVDTs located at the mid-span of the tested
specimen. The test was stopped as the actual load–deflection In order to assess the accuracy of the analytical model
behavior had declined away from the maximum test load developed, its estimated results are verified by comparing them
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1288 H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291

Table 2
Comparison of analytical model and experiment

No. Specimen designation bf/t bw/t t (mm) rc/t Moment capacity, Mcx (kNm) Mult/Mcx

Analysis (Mcx) Experiment (Mult)

1 U3.2-9 (100  40  3.2) 12.53 31.32 3.19 0.71 1.167 1.229 1.05
2 U3.2-7 (150  50  3.2) 15.47 46.72 3.20 0.70 1.523 1.380 0.91
3 U2.3-7 (150  50  2.3) 21.61 65.22 2.30 0.98 0.608 0.661 1.09
4 U2.3-9 (100  40  2.3) 17.39 43.48 2.30 0.98 0.475 0.489 1.03
5 U3.2-5 (150  50  3.2) 15.63 46.88 3.20 0.70 1.543 1.609 1.04
6 U3.2-4 (100  40  3.2) 12.34 31.25 3.20 0.70 1.155 1.178 1.02
7 U2.3-6 (150  50  2.3) 21.74 65.22 2.30 0.98 0.614 0.658 1.07
8 U2.3-7 (100  40  2.3) 17.39 43.48 2.30 0.98 0.475 0.464 0.98
9 U3.2-8 (100  50  3.2) 15.67 31.29 3.19 0.71 1.710 1.627 0.95
10 U2.3-7 (100  50  2.3) 21.74 43.48 2.30 0.98 0.675 0.632 0.94
11 U2.3-4 (100  50  2.3) 21.74 43.48 2.30 0.98 0.675 0.707 1.05
12 U1.6-9 (100  40  1.6) 25.00 62.50 1.60 1.41 0.224 0.259 1.16
13 U1.6-8 (100  50  1.6) 31.25 62.50 1.60 1.41 0.303 0.313 1.03
14 U1.6-6 (150  50  1.6) 31.92 95.77 1.57 1.43 0.278 0.330 1.19
15 U3.2-5 (100  50  3.2) 15.65 31.32 3.19 0.71 1.707 1.487 0.87

Statistical measures: (Mult/Mcx) mean ¼ 1.025; s ¼ 0.087 (S.D.).

Table 3
Comparison of analytical model and design code

No. Specimen designation bf/t bw/t t (mm) rc/t Moment capacity, Mcy (kNm) (Mcy)AISI/Mcy

Analysis (Mcy) Design code (Mcy)AISI

1 U3.2-9 (100  40  3.2) 12.53 31.32 3.19 0.71 4.813 4.996 1.04
2 U3.2-7 (150  50  3.2) 15.47 46.72 3.20 0.70 9.360 10.206 1.09
3 U2.3-7 (150  50  2.3) 21.61 65.22 2.30 0.98 4.344 4.775 1.10
4 U2.3-9 (100  40  2.3) 17.39 43.48 2.30 0.98 2.100 2.412 1.15
5 U3.2-5 (150  50  3.2) 15.63 46.88 3.20 0.70 9.490 10.283 1.08
6 U3.2-4 (100  40  3.2) 12.34 31.25 3.20 0.70 4.839 4.977 1.03
7 U2.3-6 (150  50  2.3) 21.74 65.22 2.30 0.98 4.374 4.780 1.09
8 U2.3-7 (100  40  2.3) 17.39 43.48 2.30 0.98 2.100 2.412 1.15
9 U3.2-8 (100  50  3.2) 15.67 31.29 3.19 0.71 6.192 5.452 0.88
10 U2.3-7 (100  50  2.3) 21.74 43.48 2.30 0.98 2.820 2.549 0.90
11 U2.3-4 (100  50  2.3) 21.74 43.48 2.30 0.98 2.820 2.549 0.90
12 U1.6-9 (100  40  1.6) 25.00 62.50 1.60 1.41 1.332 1.363 1.02
13 U1.6-8 (100  50  1.6) 31.25 62.50 1.60 1.41 1.767 1.410 0.80
14 U1.6-6 (150  50  1.6) 31.92 95.77 1.57 1.43 2.831 2.636 0.95
15 U3.2-5 (100  50  3.2) 15.65 31.32 3.19 0.71 6.190 5.459 0.88

Statistical measures: ((Mcy)AISI/Mcy) mean ¼ 1.004; s ¼ 0.111 (S.D.).

to the ones of experiments and the design code. It can be clearly

seen in Tables 2 and 3 that each analytical data is compared to the 1.6
individual one obtained from the experiments and the design Mult : actual moment capacity measured from
code. Fig. 13 shows the scattered deviations between the the tests
analytical and experimental data of the moment capacity about 1.4 (Mcx) : moment capacity estimated by the ana-
lytical model
the minor neutral axis (Mcx) plotted with respect to the variations
of bending flange ratio (bf/t). In the meantime, the scattered ones + 20%
1.2
between the data of analysis and design code of the moment
capacity about the major neutral axis (Mcy) are plotted with
(Mcx)
Mult

respect to the variations of bending web ratio (bw/t) in Fig. 14. 1

Both Figures indicate that the deviations of analytical data from
the experimental and design code one are still scattered within
the tolerable limits of 720%. According to a statistical analysis of 0.8
both scattered data that the results of the analytical model - 20%
developed, on average, deviate conservatively from the ones of bf : full width of the flange element
experiment and design code by 2.5% and 0.4% with the standard 0.6 t : wall thickness
deviations of 0.087 and 0.111, respectively.
Examples of implementing the method of cut-off strength in
0.4
the prediction of moment capacity are illustrated in Fig. 15. It can 0 10 20 30 40
be seen in the Figure that the analytical non-linear elastic bf
behavior tends to overestimate the actual one of the experimental
t
curve. However, the value of predicted moment capacity about the
minor neutral axis (Mcx) approaches conservatively to the Fig. 13. Degree of accuracy between the analytical model and experiments.
 ARTICLE IN PRESS

H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1289

experimental one (Mult). This is clearly indicated by the inter- found out from the predicted values of both moment capacities
secting point between the plastic mechanism and elastic curves, that the increase of their values directly correlates to the
which is relatively close to the actual-maximum moment carrying increasing dimension of the bending beam element. It is believed
capacity of the experimental curve. that this phenomenon is caused by the following arguments:
The analytical model also shows that the value of its predicted
moment capacity about both minor and major neutral axes
 The bending element is the one of the beam that directly
(Mcx and Mcy) of the investigated channel beam is significantly
carries bending stresses affected by the applied bending
affected by variations of the bending beam element. It can be
moment.
 The increase of its dimension will lead to the increase of the
1.6 second moment of area as well as the elastic section modulus
(Mcy)AISI : moment capacity calculated using the
design code about the both neutral axes of the beam.
1.4 (Mcy) : moment capacity estimated by the  Because the bending stresses are equal to the applied bending
analytical model moment divided by the elastic section modulus so that the
+ 20% increase of this modulus value will reduce the bending
1.2 stresses.
 The moment capacity is the maximum moment that can be
(Mcy)AISI

carried by the beam prior to failure and the beam is considered

(Mcy)

1 to fail as the bending stresses have fully reached the yield

strength of basic material (sy).
 Accordingly, the reduced bending stresses affected by the
0.8
- 20% increasing dimension of bending element will fully reach to the
yield strength as the applied bending moment has reached to a
bw : overall height of the web element certain value and this is in fact the same as the one of moment
0.6
t : wall thickness capacity.

0.4
0 25 50 75 100 125 On the basis of these above arguments, variations in dimension
bw of the bending element are analytically too significant in
t influencing the moment capacity of the investigated beam. In
order to illustrate the influence of this parameter, Figs. 16 and 17
Fig. 14. Degree of accuracy between the analytical model and design code. display the correlation of moment capacity and bending element

4 2
Specimen : U3.2-9 (100x40x3,2) Specimen : U2.3-7 (150x50x2.3)
(Mcx) : predicted moment capacity (Mcx) : predicted moment capacity
Bending moment, M (kNm)
Bending moment, M (kNm)

Mult : experimental moment capacity Mult : experimental moment capacity

3 Mult
Mult = 1.09
= 1.05
(Mcx) (Mcx)

2 (Mcx) 1 Elastic curve

Elastic curve Mult
Mult
1 Experimental curve Experimental curve
(Mcx)
Plastic mechanism curve Plastic mechanism curve

0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Mid-span deflection, η (mm) Mid-span deflection, η (mm)

0.8
Specimen : U1.6-9 (100x40x1.6)
(Mcx) : predicted moment capacity
Bending moment, M (kNm)

Mult : experimental moment capacity

0.6 Mult
= 1.16
(Mcx)
Elastic curve
0.4

0.2 Mult Experimental curve

Plastic mechanism curve
(Mcx)
0
0 2 4 6 8 10 12 14
Mid-span deflection, η (mm)

Fig. 15. Method of cut-off strength in the prediction of moment capacity. Specimen of nominal thickness: (a) t ¼ 3.20 mm, (b) t ¼ 2.30 mm and (c) t ¼ 1.60 mm.
 ARTICLE IN PRESS

1290 H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291

2.25 1.5
Specimen group : U3.2 Specimen group : U2.3
Mcx : moment capacity about minor Mcx : moment capacity about minor
2 1.25
neutral axis neutral axis
bf : full width of flange element bf : full width of flange element
1
Mcx (kNm)

Mcx (kNm)
1.75 t : wall thickness t : wall thickness
0.75
1.5
0.5
1.25 Experiment 0.25 Experiment
Prediction Prediction
1 0
12 13 14 15 16 15 17.5 20 22.5 25
bf bf
t t

0.75
Specimen group : U1.6
Mcx : moment capacity about minor
neutral axis
0.5 bf : full width of flange element
Mcx (kNm)

t : wall thickness

0.25

Experiment
Prediction
0
20 25 30 35
bf
t

Fig. 16. Comparison of analytical model and experiments. Specimen of nominal thickness: (a) t ¼ 3.20 mm, (b) t ¼ 2.30 mm and (c) t ¼ 1.60 mm.

21 10
Specimen group : U3.2 Specimen group : U2.3
18 Mcy : moment capacity about major Mcy : moment capacity about major
neutral axis 8
neutral axis
15 bw : overall height of web element bw : overall height of web element
Mcy (kNm)
Mcy (kNm)

12 t : wall thickness 6 t : wall thickness

9 4
6
2
3 Design Code Design Code
Prediction Prediction
0 0
30 35 40 45 50 40 45 50 55 60 65 70 75
bw bw
t t

7
Specimen group : U1.6
6 Mcy : moment capacity about major
neutral axis
5
bw : overall height of web element
Mcy (kNm)

4 t : wall thickness

3
2
1 Design Code
Prediction
0
50 60 70 80 90 100
bw
t

Fig. 17. Comparison of analytical model and design code. Specimen of nominal thickness: (a) t ¼ 3.20 mm, (b) t ¼ 2.30 mm and (c) t ¼ 1.60 mm.
 ARTICLE IN PRESS

H. Setiyono / International Journal of Mechanical Sciences 50 (2008) 1280–1291 1291

variations of the channel beam. In case of the channel beam bent capacities and this is also confirmed by the results of experiment
about the minor neutral axis, its moment capacity (Mcx) is plotted and design code.
in terms of bending flange ratio (bf/t). Meanwhile, the moment
capacity of the channel beam bent about its major neutral axis
(Mcy) is plotted in terms of bending web ratio (bw/t). Both Figs. 16 Acknowledgments
and 17 also display the verification of the analytical correlation of
moment capacity and bending element ratio using the actual one The research program presented in this paper is fully under-
measured from the experiments and the data of design code for taken by the Technology Center for Structural Strength—Agency
cold-formed steel structural members. It is clearly shown in the for the Assessment and Application of Technology (B2TKS-BPPT),
Figures that the analytical predictions well agree with the results which is located at the National Research Center for the Science
of the experiments and design code. and Technology (PUSPIPTEK) in Serpong, Tangerang, Indonesia.
The financial support to accomplish this research program is fully
granted by the Indonesian Government through a project frame-
6. Conclusions work in the Agency for the Assessment and Application of
Technology (BPPT).
An alternative-analytical model, which is based on the combined
plastic mechanism and elastic approaches, has been developed to
determine the moment capacity of a thin-walled channel steel
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adopting an energy equilibrium concept to analyze a plastic failure
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