ARTICLE IN PRESS

International Journal of Mechanical Sciences 49 (2007) 257–266

www.elsevier.com/locate/ijmecsci

Plastic mechanism and elastic–analytical approaches applied to

estimate the strength of an axially compressed-thin-walled channel steel
section beam
Harkali Setiyono,1
Technology Center for Structural Strength, Agency for the Assessment and Application of Technology, (B2TKS-BPPT), Indonesia
Received 5 November 2005; accepted 21 July 2006
Available online 3 November 2006

Abstract

This paper presents the study of an analytical model to estimate the strength of a thin-walled channel steel section beam subjected to
axial-compressive loads. The model is based on two different methods of analysis, which are performed by analysing a plastic failure
mechanism and elastic behaviour of the beam. These analytical methods can be used to establish plastic-unloading and elastic-inclining-
theoretical load-deflection behaviour of the beam. Meanwhile, the axial-compressive strength of the beam is estimated by directly
measuring the value of load at an intersection point between two different curves of the theoretical load-deflection behaviour. The
accuracy of using this analytical model is also verified by comparing its estimated data of the strength to the one obtained from a number
of tests on 38 specimens of thin-walled channel steel section under the test loads of axial compression. It is clearly shown that deviation of
the analytical data from the experimental one is still scattered within acceptable limits of 720%. A statistical analysis of the scattered
data indicates that its mean value is 1.03 with standard deviation of 0.058. This certainly means that the estimated strength, on average,
displaces from the actual one by 3% and mostly tends to be conservative.
r 2006 Elsevier Ltd. All rights reserved.

Keywords: Channel section; Local buckling; Plastic mechanisms; Effective width; Moment capacity and axial-compressive strength

1. Introduction between work done by virtual displacement of applied

loads and energy dissipating in plastic hinges of the
An analytical model of combined plastic mechanism and mechanism during deformation. The dissipated energy is
elastic approaches has been developed to estimate the determined on the basis of moment resisting capacity of
strength of a thin-walled channel steel section beam plastic hinges and their appropriate rotations.
subjected to axial-compressive loads. It has been encoun- This energy equilibrium is then analysed in more detail
tered from literature reviews that a failure process of a thin- in order to get an expression of load carrying capacity of
walled steel section under applied loads is generally the beam in term of its axial deflection. Using the
initiated by the formation of local buckling on its expression of load carrying capacity, an unloading curve
compressed elements, which subsequently can develop to of theoretical load-deflection behaviour can be produced
be local plastic failure mechanisms at collapse. In the and it is called in this paper as a plastic mechanism curve.
plastic mechanism analysis, a plastic failure mechanism of In some literatures [1,2], this curve will approximate post-
the beam affected by the axial-compressive loads is collapse behaviour of the investigated beam. The plastic
analysed according to a concept of energy equilibrium failure mechanisms of the beam can be recognized from
very careful observation on a number of thin-walled
Tel.: +62 021 7560562x1043; fax: +62 021 7560903. channel steel sections tested to failure in our laboratory.
Fig. 8 shows a type of failure mechanism of the sections
E-mail address: harkali@luk.or.id.
1
Present address: B2TKS-BPPT, Kompleks PUSPIPTEK—Serpong— and it can be seen in the figure that the plastic failure
Tangerang 15314, Indonesia. mechanism is composed of six plastic hinges, i.e., two

0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmecsci.2006.09.009
 ARTICLE IN PRESS
258 H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266

Stiffened web
Stiffened web M
element
element
Elastic stress
distribution

M : bending
Unstiffened flange moment
(a) M element (b) Unstiffened flange
M
element

Stiffened web M
element
Effective stress distribution
bew
2 Unstiffened flange
element
(c) M bew : effective width of web

Fig. 1. Effect of local buckling [3]: (a) local buckling; (b) elastic stress distribution affected by local buckling; and (c) effective width of the web element.

plastic hinges in the form of flip-disc mechanism in the web

Load
element and another four ones in both sides of the flange Elastic curve
element.
In case of elastic analysis, the effect of lgcal buckling is
taken into account in the analysis. This consideration needs Cut-off strength
to be taken because the existence of local buckling on the Pe : Elastic limit load
P Pm
affected elements of the beam causes them to be less Pe f Pf : Ultimate limit load
effective in carrying applied loads. Fig. 1 illustrates the Pm : Plastic limit load
Actual behaviour
effect of local buckling on the elastic stress distribution Plastic mechanism curve
in a compressed web element of a thin-walled channel
section subjected to bending moments [3]. On the basis of (0.0) Deflection
the illustration, an effective width concept should be used
Fig. 2. Method of cut-off strength [4–7].
to determine element widths of the investigated beam,
which are still effective in carrying the applied-compressive
loads.
Utilization of the effective element widths in the elastic 2. Analytical approach
analysis results in another formulation of load carrying
capacity in term of axial deflection. Using this formulation, The analytical approach was carried out in two stages
a theoretically inclining load-deflection behaviour of the where the first stage is a plastic analysis of a plastic
beam, which is called an elastic curve in this paper, can be mechanism model as shown in Fig. 3 and the second one is
established and this will be able to predict actual elastic an elastic analysis of the beam. Both analyses are mainly
load-deflection one. The axial-compressive strength of the aimed at developing two different formulations of plastic
investigated beam is predicted by adopting a method of and elastic load carrying capacity with respect to axial
cut-off strength [4–7] as shown in Fig. 2, where the value of deflection or axial shortening. Using these formulations,
load at the intersection of the plastic mechanism and elastic approximated load-deflection behaviour of the beam,
curves is assumed to be theoretical-axial-compressive especially in elastic and post-collapse regions, can be
strength of the beam. established and its axial-compressive strength can also be
Beyond the analytical approaches, the axial-compressive directly estimated according to the method of cut-off
strength of the beam is also experimentally assessed and strength in Fig. 2.
results obtained are used to verify the analytical predic- In order to be able to use the method of cut-off strength,
tions. In this paper, scattered deviation between analytical a computer program was written to iterate the formula-
and experimental data is limited within tolerances of tions in generating two different curves of plastic mechan-
720% and the degree of accuracy is assessed by using a ism and elastic behaviour. The iteration was performed by
statistical analysis of the scattered data populations. The firstly setting the value of axial shortening to zero and
accuracy of theoretical load-deflection behaviour in pre- incrementally increasing it until both values of plastic and
dicting the actual one is also demonstrated at the end of elastic load carrying capacity to converge in a point. The
this paper. value of load carrying capacity at this point is then
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H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266 259

Δl Formulae of Eqs. (2) and (3) are used in determining the

2 Fax reduced-plastic moment capacity of all plastic hinges
Web element
Plastic hinges throughout the plastic analysis of the mechanism model
Δl Δl
2 2
in Fig. 3.
a
If under the applied-axial-compressive loads Fax the
Ww a beam shortens by Dl, the work done by the load Fax is
Wf Δl therefore equal to the product of Fax and Dl. Meanwhile,
2 β the energy dissipating in plastic hinges is the sum of energy
Plastic hinges absorbed by the plastic hinges in the web and flange
c
Fax
yc mechanisms so that Eq. (1) may be rewritten as follows:
L
Flange element
F ax Dl ¼ ðE dis Þw þ ðE dis Þf , (4)

(Edis)w and (Edis)f are the energy absorbed by the plastic

Fig. 3. Idealized-plastic failure mechanisms of the investigated beam. hinges in the web and flange mechanisms where they are
calculated according to the sum of the product of the
reduced plastic moment capacity and the rotation angle at
each plastic hinge of both mechanisms. The energy
theoretically assumed to be the axial-compressive strength dissipated in the web and flange mechanisms can be
of the beam. obtained from the following expressions.
!
1 2a  Dl
2.1. Plastic mechanism analysis ðE dis Þw ¼ tan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½F 1  F 2 
Dlð4a  DlÞ
    
The plastic mechanism model indicated in Fig. 3 is an 0:045sy t W 2w
2
p f1 p f2
F1 ¼ ln tan þ  ln tan þ
idealization of the failure plastic mechanism of the beam in a 4 2 4 2
8 9
Fig. 8. In Fig. 3, the axial-compressive loads (Fax) are < =
aðW f  2yc Þ2 F 2ax
assumed to be applied at the center (c) so that the beam is F2 ¼ n    o ,
fully subjected to purely axial-compressive loads without :0:72s W 2 W 2 ln tan p þ f1  ln tan p þ f2 ;
y f w 4 2 4 2
additional bending. The plastic analysis of the model is ð5Þ
based on a concept of energy equilibrium, which is
formulated as follows [5–7,14]:
fð0:866sy tW 2f Þ2  ðyc F ax Þ2 gðy1 þ y2 Þ
ðE dis Þf ¼ . (6)
X
i¼n
1:73sy W 3f
E ext ¼ ½ðM 1p Þi yi , (1)
i¼1 In the formulae of Eqs. (5) and (6), parameters of f1, f2,
where Eext is the external energy and it is equal to work y1 and y2 are the inverse tangent of factors whose values
done by the virtual displacement of applied loads Fax, M1p are specified in the Appendix of this paper. Meanwhile, yc
is the reduced-plastic moment capacity of each plastic is the position of the center (c) (see Fig. 3) from the web
hinge, y is the rotation angle of each plastic hinge during element and it is expressed as follows:
deformation.
W 2f
The right-hand side term of Eq. (1) expresses the sum of yc ¼ . (7)
energy dissipation in plastic hinges and on the basis of ð2W f þ W w Þ
Murray’s formulation [1,2], the reduced-plastic moment Substituting the formulae of Eqs. (5) and (6) into the
capacity of the plastic hinge that is perpendicular to the formula of Eq. (4) and further deriving it, will end up to an
direction of applied load Fax is calculated from expression of plastic load carrying capacity (Fax)pl as
"  # follows:
1 sy bt2 F ax 2
Mp ¼ 1 , (2) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 sy bt c2 þ c22 þ 4c1 c3
ðF ax Þpl ¼ . (8)
sy is material yield strength while b and t are the length and 2c1
thickness of the plastic hinge respectively. When the plastic
The above equation is the expression of plastic load
hinge, is oriented at an angle of a to the direction of the
carrying capacity in term of axial shortening (Dl) and the
applied load Fax, the reduced-plastic moment capacity
values of c1 to c3 can be seen in Appendix.
(Mp111) becomes as follows:
Iterating the value of axial shortening (Dl) in the above
"  #
sy bt2 F ax 2 equation will generate the behaviour of unloading load-
111 1
M p ¼ M p sec a ¼ 1 sec a. (3) deflection relationship, which is called a plastic mechanism
4 sy bt
curve. In case of the value of Dl is equal to 0 (zero), the
 ARTICLE IN PRESS
260 H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266

formula of Eq. (8) becomes from

      2
p f1 p f2 t
ðF ax Þpl atðDl ¼ 0Þ ¼ ln tan þ  ln tan þ pcr ¼ 185000K , (12)
4 2 4 2 Ww
pffiffiffi
0:18sy tW f W 2w a K is a buckling constant and the constant for the stiffened
 . ð9Þ
aðW f  2yc Þ element of web (Kst) is computed using
Table 2 and Fig. 14 show that the value of (Fax)pl at 2 2 þ 4:8h Wf
(Dl ¼ 0) calculated using the above equation is quite close K st ¼ þ ; h¼ ; b ¼ ð1 þ 1:5h3 Þ0:5 . (13)
b b2 Ww
to a squash load (Fs) calculated using
The effective width of flange element (Wef) is also
F s ¼ sy tðW w þ 2W f Þ. (10)
calculated according to the same procedure as described in
the formulae of Eqs. (11) and (12) except the values of Wew
2.2. Elastic analysis and Ww in both formulae are substituted by those of Weff
and Wf. A buckling constant of the unstiffened flange
The previous section has discussed that in the elastic element (Kunst) is determined from the following equation
analysis of the beam, the effect of local buckling is and Weff obtained is then used to calculate the Wef.
necessarily considered by adopting an effective width K unst ¼ K st h2 ; W ef ¼ 0:89W eff þ 0:11W f . (14)
concept. This concept is applied to analyse a compressed
element, which is still effective in carrying applied loads. In Ramberg and Osgood have developed a formula (15) to
the investigated beam subjected to axial compressions, plot a non-linear material stress–strain curve [9]. The
both its elements of web and flanges are of compressed elastic analysis basically uses the formula (15) to estimate
ones. These elements have to be determined their effective elastic load-deflection behaviour of the investigated beam
width dimensions because in the elastic analysis, the section in this paper.
properties of the beam is determined according to its  

s 3s s ðn1Þ log 17
effective cross section instead of its full cross section. ¼ þ ; n¼1þ  7 , (15)
An effective cross section of the beam (Aef) as shown in E 7E s0:7 log ss0:85
0:7

Fig. 4 is the fundamentals of elastically developing a

formula of the load carrying capacity of the beam in term s0.7 and s0.85 are stresses corresponding to Es ¼ 0.7 E
of its axial shortening. Procedures of determining the and Es ¼ 0.85 E where E is an elastic modulus of the
effective width of web (Wew) and flange (Wef) in Fig. 4 refer basic material. These stresses are determined using stress-
to the rules as specified in Ref. [8]. The web is a stiffened strain behaviour of the basic material obtained from tensile
element and its effective width (Wew) is determined as tests.
follows: The elastic analysis of the investigated beam using the
above equation was carried out in two stages, which consist
fc
If o0:123; W ew ¼ W w , of linear-elastic analysis and non-linear elastic (inelastic)
pcr one. Both stages are aimed at developing a formula of
2 (sffiffiffiffiffiffi )4 30:2 elastic load carrying capacity (Fax)e in term of axial
f f
If c X0:123; W ew ¼ W w 41 þ 14 c
 0:35 5 , shortening (Dl) relationship. In the first stage, the first
pcr pcr term in the right-hand side of Eq. (15), which relates to
ð11Þ Hooke’s law, is further analysed to get the following linear-
elastic load-deflection relationship:
fc is a compressive stress in the effective element and it can
be equated to design strength (py) or yield strength (sy). EtðW ew þ 2W ef ÞDl
ðF ax ÞeðlinearÞ ¼ . (16)
Meanwhile, pcr is a local buckling stress and calculated L

Δl Wew Wew
Stiffened web
Fax Δl
2 Effective-stiffened 2 2
element 2
web element
Δl
Δl
2 Ww Ww
2

L Wef c L
Wf c Wf

Fax Unstiffened Effective-unstiffened

(a) flange element (b) flange element

Fig. 4. Full and effective cross sections of the investigated beam: (a) full cross section and (b) effective cross section.
 ARTICLE IN PRESS
H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266 261

In the meantime, the non-linear elastic load-deflec- testing machine RME 100 Schenck Trebel whose maximum
tion one is obtained from analysing the second term of capacity is 100 kN.
the right-hand side of Eq. (15) to get its formulation as The tensile specimens were tested in a room temperature
follows: to fracture and during the tests, a relationship of static-
tensile load to specimen deformation was always mon-
ðF ax ÞeðnonlinearÞ ¼ ðDF ax Þinel þ F 0:7 ;
itored by means of extensometer that was mounted at a
F 0:7 ¼ s0:7 tðW ew þ 2W ef Þ, gauge length of 100 mm. The mechanical properties of the
" #1=n basic material can be identified by evaluating the tensile
7Esðn1Þ n
0:7 ftðW ew þ 2W ef Þg ðDl  Dl 0:7 Þ
ðDF ax Þinel ¼ , test load-deformation behaviour obtained and their values,
3L on average, are as follows [10]:
ð17Þ
Dl0.7 is axial shortening that corresponds to F0.7 and Eqs.  ultimate tensile strength (sUTS) ¼ 322.25 MPa,
(8), (16) and (17) are iterated using the written computer  yield strength (sy) ¼ 191.50 MPa,
program to implement the method of cut-off strength in  modulus of elasticity (E) ¼ 196.45  103 MPa,
estimating the axial-compressive strength of the investi-  stress corresponding to Es ¼ 0.7 E, (s0.7) ¼ 134.33 MPa,
gated beam.  stress corresponding to Es ¼ 0.85 E, (s0.85) ¼ 115.67 MPa.

3. Experimental investigation A subsequent step of the experimental investigation is

axial-compressive tests on 38 specimens of thin-walled
In the experimental investigation, tensile tests were channel steel section beam. The specimen is cold-formed
initially performed to assess mechanical properties of the from the carbon steel sheet JIS G 3141-SPCC as above
basic material used to manufacture the thin-walled channel mentioned and its detail design can be seen in Fig. 6. The
steel section beam. The basic material is of a carbon steel axial-compressive tests were also performed in an ambient
sheet of Standard JIS G 3141-SPCC and tensile test temperature using the testing machine RME 100 Schenck
specimens are designed according to Standard JIS Z 2201 Trebel of a 100 kN maximum capacity until the specimen
no. 13A (see Fig. 5). The tensile tests were conducted on a beams completely failed. In order to ascertain that the

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

Nominal thickness t = 1.00 – 3.20 mm

Fig. 5. Design of tensile test specimen [11,12].

bw Nominal dimensions:
bw = 100 and 150 mm
Ww bf = 40 and 50 mm
t = 1.00, 1.60 and
2.30 mm
r
r = 2.25mm
bf Wf 90°° 90° t L = 400, 500 and
L
600 mm
Ww = bw – t
Wf = bf – 0.5t

Fig. 6. Design of axially compressed-tested specimen [10].

 ARTICLE IN PRESS
262 H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266

Fig. 8. Local-plastic failure mechanisms of the tested beam [10].

Fig. 7. Configuration of axially compressive tests [10].

Table 1
Comparison of theoretical and actual strength

Specimen designation bf bw L (mm) Axial-compressive strength (kN) F ex

t t F th
Fth (theory) Fex (experiment)

U1-1(100  40  1) 40.82 101.82 400 16.06 16.00 0.99E1.00

U1-6(100  40  1) 42.93 106.11 500 14.62 15.00 1.03
U1-7(100  40  1) 40.87 101.96 600 15.63 16.40 1.05
U1-3(100  50  1) 51.16 101.98 400 15.43 17.15 1.11
U1-4(100  50  1) 50.20 99.90 500 15.81 17.20 1.09
U1-9(100  50  1) 48.82 98.12 599 16.17 17.80 1.10
U1-3(150  50  1) 51.57 154.64 400 18.02 17.20 0.95
U1-4(150  50  1) 50.06 149.70 500 18.79 18.40 0.98
U1-7(150  50  1) 51.04 152.86 600 17.86 18.70 1.05
U1-3(100  40  1) 40.06 99.94 400 16.61 16.80 1.01
U1-5(100  40  1) 40.76 101.22 500 15.83 16.30 1.03
U1-4(100  40  1) 41.56 104.06 600 15.10 17.20 1.14
U1-5(100  50  1) 50.00 99.90 500 15.79 15.50 0.98
U1-7(100  50  1) 50.01 99.68 600 15.62 17.30 1.11
U1-1(150  50  1) 50.14 150.15 400 19.00 18.30 0.96
U1-1(100  50  1) 53.00 106.21 400 14.35 16.50 1.15
U1-8(150  50  1) 49.80 150.00 600 18.49 17.00 0.92
U1.6-5(100  40  1.6) 24.88 62.50 500 38.33 37.50 0.98
U1.6-7(100  40  1.6) 25.33 63.27 600 37.53 37.20 0.99E1.00
U1.6-4(100  40  1.6) 25.13 62.50 500 38.21 37.50 0.98
U1.6-2(100  40  1.6) 25.35 63.38 400 38.10 38.50 1.01
U1.6-7(100  50  1.6) 31.65 63.35 600 35.29 36.00 1.02
U1.6-4(100  50  1.6) 31.13 62.73 500 36.68 37.00 1.01
U1.6-1(100  50  1.6) 32.53 64.96 400 34.53 38.00 1.10
U1.6-8(150  50  1.6) 32.93 98.62 600 39.11 37.50 0.96
U1.6-4(150  50  1.6) 33.36 100.00 500 38.87 39.25 1.01
U2.3-6(100  40  2.3) 17.52 43.82 500 63.71 71.00 1.11
U2.3-5(100  40  2.3) 17.53 43.30 500 64.44 66.75 1.04
U2.3-4(150  50  2.3) 22.66 68.14 500 77.26 83.50 1.08
U2.3-5(150  50  2.3) 22.65 68.14 500 77.25 83.00 1.07
U2.3-2(100  50  2.3) 21.57 43.43 400 67.26 71.75 1.07
U2.3-3(100  40  2.3) 17.30 43.50 400 66.39 71.25 1.07
U2.3-2(150  50  2.3) 21.70 64.96 400 84.51 84.50 1.00
U2.3-3(150  50  2.3) 21.24 64.04 400 84.64 80.00 0.95
U2.3-1(150  50  2.3) 22.03 65.90 401 82.55 83.75 1.01
U2.3-8(100  40  2.3) 17.46 43.80 400 65.70 72.50 1.10
U2.3-8(150  50  2.3) 21.90 65.79 600 81.19 84.30 1.04
U2.3-9(150  50  2.3) 21.88 65.75 600 81.50 80.00 0.98
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H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266 263

specimen beams are really subjected to axial-compressive According to a statistical analysis of the scattered
test loads, they are located in-between the upper and lower deviation data, its mean value is 1.03 with the standard
clamping devices in such away that the centre line of the deviation of 0.058. These statistical measures mean that the
devices exactly coincides with the longitudinal centre line of average estimated strength tends to underestimate the actual
the specimens. one by 3% and this is of course a considerably safe
This test arrangement is shown in Fig. 7 and it is clearly prediction. Figs. 11–13 show theoretical load-deflection
seen that the LVDTs are used to measure axial shortening behaviour of the investigated beam and this is represented
as well as lateral deflection of the web element. Actual by the plastic mechanism and elastic curves. The plastic
behaviour of load-deflection relationship is always mon- mechanism curve is established by iterating Eq. (8), whereas
itored during the tests and plotted in X–Y recorders. The the Eqs. (16) and (17) are iterated to establish the elastic one.
axial-compressive strength of the tested specimens is On the comparison of the theoretical load-deflection
measured from a load indicating device of the testing curve to the experimental one, it can be seen that the actual
machine and also from a peak value of test load in the elastic behaviour is well predicted by the one obtained from
actual load-deflection behaviour obtained. A mode of
failure at each specimen is carefully observed and this
repeatedly occurs in the form of locally plastic failure 1.6
Fex : experimental-axial-compressive strength
mechanisms as indicated in Fig. 8. Fth : theoretical-axial-compressive strength
bw : overall flange width
1.4 t : wall thickness

+ 20%
4. Verification of analytical results 1.2

The axial-compressive strength of the beam estimated Fex

using the analytical approach developed in this research Fth 1

program is verified by comparing it to actual data

measured in the tests. Table 1 indicates the comparison 0.8
of individual data of estimated and actual strength of 38 - 20%
thin-walled channel steel section beams axial-compressively
0.6
loaded. The ratio of individual-experimental-axial-com-
pressive strength (Fex) and theoretical one (Fth) is also
plotted in terms of web ratio (Ww/t) and flange ratio (Wf/t) 0.4
10 20 30 40 50 60
as shown in Figs. 9 and 10. It is clearly seen in the figures
bf
that this ratio data, which also expresses the percentage of t
deviation between estimated and actual strength, still lies
within an acceptable limits of 720% and mostly scatters in Fig. 10. Scattered deviation of theoretical and actual strength comparison
in term of the flange ratio.
the conservative region (1.00p(Fex/Fth)p1.20).

1.6 35
Fex : experimental-axial-compressive strength Specimen U1-1(100x40x1)
Fth : theoretical-axial-compressive strength
bw : overall web height 30
1.4 t : wall thickness Fth : Theoretical ultimate axial load
Fex : Experimental ultimate axial load
Fex / Fth = 0.99 ≈ 1.00
Axial-compressive load (kN)

+ 20% 25
1.2

20
Fex
Fth

1 Fex Elastic curve

15
Fth
0.8
- 20%
10
Experimental curve
0.6
5

0.4 Plastic mechanism curve

0 25 50 75 100 125 150 175 200 0
bw 0 1 2 3 4
t Axial shortening of beam (mm)

Fig. 9. Scattered deviation of theoretical and actual strength comparison Fig. 11. Theoretical and actual load-deflection behaviour (nominal
in term of the web ratio. t ¼ 1.00 mm).
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264 H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266

70 100
Specimen: U1.6-5 (100x40x1.6)
Elastic curve
90
60 Fth : Theoretical ultimate axial load
Fex : Experimental ultimate axial load 80 Fth
Fex / Fth = 0.98
Fex
Axial-compressive load (kN)

Axial-compressive load (kN)

50 70
Fth
Elastic curve 60
40 Experimental curve
Fex
50
30 Experimental curve Plastic mechanism curve
40

20 Plastic mechanism curve 30 Specimen: U2.3-2 (150x50x2.3)

20
Fth : Theoretical ultimate axial load
10 Fex : Experimental ultimate axial load
10 Fex / Fth = 1.00

0 0
0 1 2 3 4 5 0 1 2 3 4 5
Axial shortening of beam (mm) Axial shortening of beam (mm)

Fig. 12. Theoretical and actual load-deflection behaviour (nominal Fig. 13. Theoretical and actual load-deflection behaviour (nominal
t ¼ 1.60 mm). t ¼ 2.30 mm).

Table 2
Comparison of Eqs. (9) and (10)

Specimen designation bf bw L (mm) Plastically axial load carrying capacity at Dl ¼ 0 (kN) ðF ax Þpl
t t Fs
(Fax)pl of Eq. (9) Fs of Eq. (10)

U1-1(100  40  1) 40.82 101.82 400 34.05 33.37 1.02

U1-6(100  40  1) 42.93 106.11 500 32.80 32.14 1.02
U1-7(100  40  1) 40.87 101.96 600 34.10 33.42 1.02
U1-3(100  50  1) 51.16 101.98 400 37.95 37.21 1.02
U1-4(100  50  1) 50.20 99.90 500 38.73 37.97 1.02
U1-9(100  50  1) 48.82 98.12 599 39.37 38.61 1.02
U1-3(150  50  1) 51.57 154.64 400 47.00 46.09 1.02
U1-4(150  50  1) 50.06 149.70 500 48.40 47.46 1.02
U1-7(150  50  1) 51.04 152.86 600 47.44 46.52 1.02
U1-3(100  40  1) 40.06 99.94 400 34.80 34.10 1.02
U1-5(100  40  1) 40.76 101.22 500 33.92 33.24 1.02
U1-4(100  40  1) 41.56 104.06 600 33.35 32.68 1.02
U1-5(100  50  1) 50.00 99.90 500 38.65 37.90 1.02
U1-7(100  50  1) 50.01 99.68 600 38.61 37.86 1.02
U1-1(150  50  1) 50.14 150.15 400 48.52 47.57 1.02
U1-1(100  50  1) 53.00 106.21 400 38.65 37.90 1.02
U1-8(150  50  1) 49.80 150.00 600 48.36 47.42 1.02
U1.6-5(100  40  1.6) 24.88 62.50 500 55.16 54.05 1.02
U1.6-7(100  40  1.6) 25.33 63.27 600 54.61 53.51 1.02
U1.6-4(100  40  1.6) 25.13 62.50 500 55.41 54.29 1.02
U1.6-2(100  40  1.6) 25.35 63.38 400 54.69 53.59 1.02
U1.6-7(100  50  1.6) 31.65 63.35 600 60.77 59.59 1.02
U1.6-4(100  50  1.6) 31.13 62.73 500 61.49 60.29 1.02
U1.6-1(100  50  1.6) 32.53 64.96 400 59.30 58.14 1.02
U1.6-8(150  50  1.6) 32.93 98.62 600 73.32 71.89 1.02
U1.6-4(150  50  1.6) 33.36 100.00 500 72.39 70.97 1.02
U2.3-6(100  40  2.3) 17.52 43.82 500 78.09 76.53 1.02
U2.3-5(100  40  2.3) 17.53 43.30 500 78.95 77.36 1.02
U2.3-4(150  50  2.3) 22.66 68.14 500 105.36 103.30 1.02
U2.3-5(150  50  2.3) 22.65 68.14 500 105.34 103.28 1.02
U2.3-2(100  50  2.3) 21.57 43.43 400 87.38 85.68 1.02
U2.3-3(100  40  2.3) 17.30 43.50 400 78.69 77.10 1.02
U2.3-2(150  50  2.3) 21.70 64.96 400 109.88 107.73 1.02
U2.3-3(150  50  2.3) 21.24 64.04 400 108.04 107.91 1.02
U2.3-1(150  50  2.3) 22.03 65.90 401 108.65 106.53 1.02
U2.3-8(100  40  2.3) 17.46 43.80 400 77.94 76.37 1.02
U2.3-8(150  50  2.3) 21.90 65.79 600 109.25 107.11 1.02
U2.3-9(150  50  2.3) 21.88 65.75 600 109.15 107.02 1.02
 ARTICLE IN PRESS
H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266 265

120 applied to the analysis of an idealized plastic failure

mechanism model of the beam. In the elastic approach, the
100 effect of local buckling on the compressed elements is taken
into account by adopting an effective width concept in
determining the cross section of the beam, which is still
80
+ 20% effective to carry applied-compressive loads. The strength
of the investigated beam is estimated by implementing the
Fs (kN)

60 method of cut-off strength on the two different curves of

- 20%
plastic and elastic load carrying capacity.
40 The accuracy of using the method developed is also
assessed by comparing its predicted results to actual ones
20
(Fax)pl at Δl = 0 : plastically axial-compressive load measured in axial compression tests of thin-walled channel
calculated using equation (9)
steel section beam specimens. The assessment has indicated
Fs : squash load calculated using equation (10)
that the analytical model presented in this paper can
0
0 20 40 60 80 100 120
predict the axial-compressive strength very quite well and
(Fax)pl at Δl = 0 (kN) tends to underestimate the actual strength by 3%. The
analytical model also shows to be able to estimate load-
Fig. 14. Comparison of Eqs. (9) and (10). deflection behaviour of the beam, which is close to the
actual behaviour displayed from the test results.

the elastic-analytical approach. Meanwhile, the post-

collapse behaviour of the beam can be predicted by the Acknowledgements
plastic mechanism (unloading) curve with fairly accurate
and the prediction tends to underestimate it. The reasons of The research program reported herein is financed by the
this underestimated prediction can be explained as follows: Indonesian Government through the project framework in
the Agency for the Assessment and Application of
 The test results show that the local-plastic failure Technology (BPPT). Valuable contributions rendered by
mechanism of the beam is not always formed exactly engineers and technicians in the Center for the Technology
at its mid-span as assumed in the analytical approach. of Structural Strength (B2TKS-BPPT) especially in the
 The effect of strain hardening is not taken into account experimental investigation are gratefully acknowledged.
in the analysis at the plastic hinge zones and the material The author also thanks P.T. Duta Laserindo Metal—Sheet
is assumed to follow the elastic-perfectly-plastic stress– Metal Job in Bekasi—Indonesia for manufacturing a
strain behaviour. number of specimens used to support the research
activities.
As stated in the method of cut-off strength, Figs. 11–13
also indicate the analytical-axial-compressive strength to be
determined as the inelastic behaviour curve has been
intercepted by the plastic mechanism one. The maximum Appendix
plastic load carrying capacity indicated by the value at the
intersection between the plastic mechanism curve and the The following is detail explanation of the factors
vertical axis is equal to a squash load. A calculation of this introduced in Eq. (8).
load using either Eq. (9) or (10) produces similar values as  c11 c12 c16
can clearly be seen in Table 2 and plotted in Fig. 14. In this c1 ¼ þ ,
c13 ðc14  c15 Þ c17
figure, the different data calculated using both equations
are scattered quite closely to the solid diagonal line and this 
indicates that the ratio of Eqs. (9) and (10) data is nearly c2 ¼ Dl,
unity. Thus, it can be certainly found out that both Eqs. (9)
and (10) can actually be used to determine the value of  c32 ðy1 þ y2 Þ
squash load. c3 ¼ c11 xc31 ðc4  c15 Þ þ ,
c17

5. Conclusions where
!
A combined method of plastic mechanism and elastic 2a  Dl
c11 ¼ 2tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; c12 ¼ aðW f  2yc Þ2 ;
approaches has been developed to analyse the strength Dlð4a  DlÞ
of a thin-walled channel steel section beam subjected to
W 2f
axial-compressive loads. The plastic mechanism approach yc ¼ ,
is performed on the basis of an energy equilibrium concept ð2W f þ W w Þ
 ARTICLE IN PRESS
266 H. Setiyono / International Journal of Mechanical Sciences 49 (2007) 257–266

 
p f1 0:65W f L
c13 ¼ 0:72sy W 2f W 2w ; c14 ¼ ln tan þ ; AB ¼ 0:866W f ; BB11 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
4 2 1:5L  1:16 Dlð4a  DlÞ
  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p f2 0:5L þ 0:77 Dlð4a  DlÞ
c15 ¼ ln tan þ , cos b1 ¼ .
4 2 L
0:045sy t2 W 2w
c16 ¼ y2c ðy1 þ y2 Þ; c17 ¼ 1:73sy W 3f ; c31 ¼ ,
a References
0 1 [1] Murray NW, Khoo PS. Some basic plastic mechanisms in the local
B 5:66a C buckling of thin-walled steel structures. International Journal of
c32 ¼ ð0:866sy þ W 2f Þ2 ; f1 ¼ tan1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; Mechanical Sciences 1981;23(12):703–13.
2 2
4W w  32a [2] Murray NW. Introduction to the theory of thin-walled structures.
0 1 Oxford Engineering Sciences Series 1986;13:313.
[3] Rhodes J. Design of cold-formed steel members. Elsevier Science
B 5:66a C Publishers Ltd.; 1991.
f2 ¼ tan1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA,
2 2 [4] Bakker MCM. Yield line analysis of post-collapse behaviour of thin-
4W w  32a walled steel members. Heron 1990;35(3):3–50.
[5] Harkali Setiyono, Web crippling of cold-formed plain channel steel
0qffiffiffiffiffiffiffiffiffiffiffiffiffi1
section beams, PhD thesis, University of Strathclyde, August, 1994,
1  x21 pp. 199–234.
y1 ¼ tan1 @ A;
x1 [6] Harkali Setiyono, Application of plastic mechanism approach to
analyse the strength of a light gauge steel section, Proceedings—the
ðA1 B1 Þ2 þ ðA1 C 1 Þ2  ðBB1 Þ2  ðBC 1 Þ2 þ 2ðBB1 ÞðBC 1 Þ cos b2 10th international pacific conference on automotive engineering
x1 ¼ . (IPC-10), Melbourne—Australia, May 23–28, 1999. p. 53–8.
2ðA1 B1 ÞðA1 C 1 Þ
[7] Harkali Setiyono and Abdul Rachman, A plastic mechanism
The factor (a) is half width of the web flip-disc mechanisms approach in an analytical model to estimate the strength of a
as shown in Fig. 3. lightweight steel structure, Proceedings—21st conference of ASEAN
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi federation of engineering organizations (CAFEO—21), Yogyakarta,
22–23 October, 2003. p. 1/15–15/15.
0:43W f AD 3W 2f  AD2
[8] British Standard Institution, Structural use of steelwork in building,
A1 B1 ¼ ;
1:5W 2f  AD2 Part 5. Code of practice for design of cold-formed sections, BS 5950,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1987.
[9] Lau SCW, Hancock GJ. Inelastic buckling analyses of beams,
0:75L  1:16 Dlð4a  DlÞ
AD ¼ W f , columns and plates using the spline finite strip method. Thin-Walled
L Structures 1989;7:213–38.
[10] Harkali Setiyono, The Development of an analytical model for the
0:65W f L strength of light gauge steel structures, Final research report, Agency
A1 C 1 ¼ 0:75W f ; BB1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; for the Assessment and Application of Technology (BPPT), 2001,
0:75L þ 1:16 Dlð4a  DlÞ
(Translated from Indonesian language).
BC 1 ¼ 0:866W f , [11] Japanese Standards Association, JIS handbook—ferrous materials &
metallurgy I, 1995. p. 34.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [12] Japanese Standards Association, JIS handbook—ferrous materials &
L  0:77Dlð4a  DlÞ
cos b2 ¼ , metallurgy II, 1995. p. 253–72.
L [14] Harkali Setiyono, Djoko W. Karmiadji, A. Anton, The Development
0qffiffiffiffiffiffiffiffiffiffiffiffiffi1 of a combined plastic mechanism and elastic–analytical model to
1  x22 estimate the moment capacity of a thin-walled channel section
y2 ¼ tan1 @ A; (IPC2001D081), Proceedings (CD-ROM)—the 11th International
x2 Pacific Conference on Automotive Engineering (IPC-11), Shianghai-
China, November 8–9th, 2001.
ðA11 C 11 Þ2 þ ðB11 C 11 Þ2  ðABÞ2  ðBB11 Þ2 þ 2ðABÞðBB11 Þ cos b1
x2 ¼ ,
2ðA11 C 11 ÞðB11 C 11 Þ

A11 C 11 ¼ 0:75W f ; Further reading

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
0:86W f 0:87L Dlð4a  DlÞ  0:34Dlð4a  DlÞ [13] Mahendran M. Local plastic mechanisms in thin steel plates under in-
B11 C 11 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; plane compression. Thin-Walled Structures 1997;27(3):245–61.
1:5L  1:16 Dlð4a  DlÞ