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Lateral–torsional buckling of steel web tapered tee-section cantilevers
Journal of Constructional Steel Research 87 (2013) 31–37
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Journal of Constructional Steel Research
Lateral–torsional buckling of steel web tapered tee-section cantilevers Wei-bin Yuan a, Boksun Kim b,⁎, Chang-yi Chen a a b
College of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou, China School of Marine Science and Engineering, Plymouth University, Plymouth, PL4 8AA, UK
a r t i c l e
i n f o
Article history: Received 17 January 2013 Accepted 29 March 2013 Available online 11 May 2013 Keywords: Lateral–torsional buckling Web tapered tee sections Tapered cantilevers Analytical study Finite element analysis
a b s t r a c t In this paper an analytical model is presented to describe the lateral–torsional buckling behaviour of steel web tapered tee-section cantilevers when subjected to a uniformly distributed load and/or a concentrated load at the free end. To validate the present analytical solutions finite element analyses using ANSYS software are also presented. Good agreement between the analytical and numerical solutions is demonstrated. Using the present analytical solutions, the interactive buckling of the tip point and uniformly distributed loads is investigated and a parametric study is carried out to examine the influence of section dimensions on the critical buckling loads. It is found that web tapering can increase or decrease the critical lateral–torsional buckling loads, depending on the flange width of the beam. For a beam with a wide flange (width/depth = 0.96) the critical buckling load is increased by 2% by web tapering, whereas for a beam with a narrow flange (width/depth = 0.19) web tapering reduces the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Tee-section beams are widely used in modern construction due to their structural efficiency. The main feature of a tee-section beam is the monosymmetry of the cross-section. For most tee-section cantilever beams carrying gravity loading the flange is positioned at the top, in which case the flange is in tension and the unstiffened portion of the web is in compression. Because the neutral axis of a tee-section beam is closer to the flange the maximum compressive stress in the web is much higher than the maximum tensile stress in the flange. This means that such beams fail by compressive stress and the lateral–torsional or lateral–distortional buckling could be one of the main failure modes [1]. The instability of monosymmetric I-beams under various loading conditions has been studied by many researchers [2–8]. The main difficulty of the problem is the presence of an additional torque, owing to the monosymmetry of the section, arising from the pre-buckling longitudinal bending stresses as the beam twists during the buckling. This additional torque causes an effective change in the torsional stiffness of the beam. This feature does not exist in symmetric beams, and was not addressed until the 1940s [9]. Since then different modifications of the torsional stiffness to account for the effect of the additional torque have been proposed [6]. Steel web tapered tee-section beams are very popular because of their aesthetic features and light weight. These beams are mainly cantilevered and have the advantage of low weight-to-strength ratios. They are structurally efficient since the web can be tapered along the beam to closely match the variation of the bending moment of the beam. The depth of ⁎ Corresponding author. Tel.: +44 1752 586135; fax: +44 1752 586101. E-mail address: [email protected] (B. Kim). 0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcsr.2013.03.026
the beam is largest at the fixed support, where its bending moment is greatest, and gradually decreases towards the free end. Although steel web tapered tee-section cantilevers are commonly used, research into the instability of such beams is very limited. The majority of the existing literature deals with the lateral–torsional buckling of tapered I-beams [10–18]. Studies into tapered tee-section cantilevers are few. One rare example is by Fischer and Smida [19]. Kitipornchai and Trahair [10] derived differential equations for the non-uniform torsion of tapered I-beams by analyzing the deformations of the flanges and investigated the elastic flexural–torsional buckling of simply supported tapered I-beams. Later, they extended their method to tapered mono-symmetric I-beams. [11]. Yang and Yau [13], and Bradford and Cuk [14] presented numerical investigations on the lateral–torsional and lateral–distortional buckling of tapered monosymmetric I-beams using finite element methods. Studies by Andrade et al. [16,17] have shown that the lateral torsional buckling loads of simply supported web tapered I-beams were decreased by as much as 20 to 40% as the degree of taper increased. This disagrees with the earlier work by Kitipornchai and Trahair [10], which concluded that the critical loads of web tapered beams did not vary greatly as the degree of taper increased since the torsional stiffness was insensitive to the degree of taper. The disagreement could be because a very short beam of 1.52 m span was used in the work of Kitipornchai and Trahair [10], while Andrade et al. [16,17] used long beams of 6 m, 9 m, and 12 m. The boundary conditions of a beam seem to influence the lateral torsional buckling loads of web tapered I-beams. The buckling loads are decreased for simply supported beams [16–18], while being increased for fix-end beams [18] and cantilevers [16–18], compared with those of un-tapered ones. The increase for the cantilevers is significant and increases as the degree of taper increases. However those three studies
32
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
[16–18] are based on analytical and numerical analyses and hence the results should be validated by experimental work. Web tapered tee-section cantilevers may behave differently from tapered I-beam cantilevers. The lack of a bottom flange means that the lower part of the web is in compression and this increases buckling instability. A study by Fisher and Smida [19] discussed the failure modes of such beams, however their experimental results were not compared to any un-tapered beams. Furthermore no analytical study has been carried out on this topic. In this paper an analytical model is presented to describe the lateral– torsional buckling behaviour of steel web tapered tee-section beams when subjected to a uniformly distributed load and/or a concentrated load at the free end. To validate the present analytical solutions finite element analyses using ANSYS software are also presented. Using the present analytical solutions, the interactive buckling of the distributed and concentrated loads is discussed and a parametric study is carried out to provide the optimum design of tee-section beams against lateral–torsional buckling. 2. Lateral–torsional buckling analysis of steel web tapered tee-section cantilevers Consider a web tapered tee-section cantilever subject to a uniformly distributed load and a concentrated load at its free end, as shown in Fig. 1. Let x be the longitudinal axis of the beam, y and z be the cross-sectional axes parallel to the web and flange, respectively. For convenience, the origin of coordinates was chosen to be the centroid of the section. Due to the tapering of the web, the section properties of the beam are a function of the coordinate x and can be expressed as follows: 2
y¼
Iy ¼
bf t f 2
þ t w ðbwo −x tan α Þ t f þ bwo −x2 tan α bf t f þ t w ðbwo −x tan α Þ
t f b3f ðbwo −x tan α Þt 3w þ 12 12
ð1Þ
and z-axes, respectively, J is the torsional constant of the section, bf is the flange width, tf is the flange thickness, bwo is the web depth at the support (x = 0), tw is the web thickness, and α is the tapering angle. Assume that when lateral–torsional buckling occurs, the displacements of the beam can be described as follows: vðxÞ ¼ ∑ An
xnþ2 xnþ2 l
ð6Þ
xnþ1 l
ð7Þ
wðxÞ ¼ ∑ Bn n¼0
ϕðxÞ ¼ ∑ C n n¼0
where v and w are the transverse and lateral displacements of the beam defined at the shear centre, respectively, ϕ is the angle of rotation of the cross-section, An, Bn and Cn (n = 0, 1, 2, …) are the constants to be determined, and l is the length of the beam. Note that the displacement functions assumed in Eqs. (5)–(7) satisfy the clamped boundary conditions (v = w = ϕ = 0 and dv/dx = dw/dx = 0) at the support (x = 0). The strain energy of a tee-section beam due to the buckling displacements can be calculated using the following formula [20]: l
"
EIz U¼∫ 2 o
d2 v dx2
ð2Þ
" 2 # ðbwo −x tan α Þ2 b −x tan α þ t f −y þ wo 2 12 " 2 2 # tf tf ð3Þ þ y− þ bf t f 12 2
EIy þ 2
!2 # d2 w GJ dϕ 2 þ dx 2 dx dx2
ð8Þ
where E is the Young's modulus and G is the shear modulus. Note that for a tee-section the warping constant is zero and thus no warping energy is involved in Eq. (8). Substituting Eqs. (5)–(7) into Eq. (8) yields l xn 2 xn 2 EI y EI z ∑ An ðn þ 1Þðn þ 2Þ dx þ ∫ 4 ∑ Bn ðn þ 1Þðn þ 2Þ dx 4 l l n¼0 n¼0 o 2l o 2l l xn 2 GJ þ∫ 2 ∑ C n ðn þ 1Þ dx: ð9Þ l 2l n¼0 o l
The loss of the potential energy of the externally applied loads due to the buckling displacements can be calculated using the following formula [2,4,6,20]: l
3
!2
U¼∫
I z ¼ t w ðbwo −x tan α Þ
W ¼ −∫ M z ϕ
3
bf t f þ ðbwo −x tanα Þt w J¼ 3
ð5Þ
l
n¼0
ð4Þ
where y is the distance from the top of the section to the neutral axis, Iy and Iz are the second moments of the cross-sectional area about the y-
o
2 l d2 w 1 dϕ dx− ∫ Mz βz dx 2 2o dx dx
ð10Þ
where Mz is the internal bending moment about the z-axis of the beam in the pre-buckling stage, which is generated due to the externally applied loads, and βz is the parameter describing the monosymmetric property
Fig. 1. A web tapered tee-section cantilever beam subject to a uniformly distributed load and a concentrated load at its free end.
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
33
of the section [2,4,6]. For a beam subject to a uniformly distributed load and a concentrated load at its free end Mz can be expressed as follows:
where Π = U + W is the total potential energy. Substituting Eqs. (9) and (13) into Eq. (14) yields
q 2 Mz ðxÞ ¼ P ðl−xÞ þ ðl−xÞ 2
ðm þ 1Þðm þ 2Þ∑ An ðn þ 1Þðn þ 2Þ∫
ð11Þ
n¼0
where P an q are the concentrated and uniformly distributed loads respectively, shown in Fig. 1. The monosymmetric property parameter is defined as [2,4,6]
βz ¼ ¼
1 Iz
∫ z ydA þ ∫ y dA −2y0
1 Iz
∫ z ydA þ ∫ y dA
A 2
A
! 3
A
EIy xmþn dx 4 l o l
n¼0
l
ðm þ 1Þ∑ C n ðn þ 1Þ∫ n¼0
4 4 tf b t þ w y−t f − bw −y þ t f ¼ y−t f − w þ y− 12I z 2 12I z 2 4I z bf tf 4 4 þ y − y−t f þ 2 y− ð12Þ 4I z 2 bw t 3w
3 t f bf
where bw = bwo–xtanα and y0 is the distance between the shear centre and the centroid, and positive as the convention is upwards, as shown in Fig. 1. Substituting Eqs. (6) and (7) into Eq. (10) yields
ð15Þ
l
n¼0
tf þ 2 y− 2
EIz xmþn dx ¼ 0 l4 l
ðm þ 1Þðm þ 2Þ∑ Bn ðn þ 1Þðn þ 2Þ∫ ¼ ðm þ 1Þðm þ 2Þ∑ C n ∫
3
A
o
l
! 2
l
o
o
Mz xmþnþ1 dx l2 l
ð16Þ
GJ xmþn M xmþnþ1 dx ¼ ∑ Bn ðn þ 1Þðn þ 2Þ∫ 2z dx l l2 l n¼0 o l l
l
þ ðm þ 1Þ∑ C n ðn þ 1Þ∫ n¼0
o
M z β z xmþn dx l l2
ð17Þ
where m = 0, 1, … represents the number of equations. Eqs. (15)–(17) can be further written as follows: Am ¼ 0
ð18Þ 1
ðm þ 1Þðm þ 2Þ∑ ðn þ 1Þðn þ 2Þα mn Bn n¼0
3
¼ λðm þ 1Þðm þ 2Þ∑ α mn C n
xnþ1 xn M ∑ Bn ðn þ 1Þðn þ 2Þ dx W ¼ −∫ 2z ∑ C n l l n¼0 n¼0 o l l
ð13Þ
xn i2 1 M β − ∫ z2 z ∑ C n ðn þ 1Þ dx: 2 o l n¼0 l h
l
2
n¼0
4
ð14Þ
ðn ¼ 0; 1; …Þ
l
1
α mn ¼ ∫ o
¼
α mn ¼ ∫
l
α mn ¼ ∫ o
bf = 100 mm
! 3 1 t w l tan α 1 − ðm þ n þ 1Þ 12Iyo ðm þ n þ 2Þ
GJ xmþn GJ dx ¼ o l l2 l
ð21Þ
! 3 1 t l tan α 1 − w ð22Þ ðm þ n þ 1Þ 3J o ðm þ n þ 2Þ
M z xmþnþ1 P dx ¼ ðm þ n þ 2Þðm þ n þ 3Þ l2 l
þ
bf = 50 mm
1.4
βz(x) / βz(0)
l3 l
2
EI y xmþn dx l4 l
EI yo
o
bf = 250 mm
ð20Þ
where m = 0, 1, … represents the number of equations and λ is the k loading factor and αmn (k = 1, 2, 3, 4) are defined as follows:
3
1.6
n¼0
n¼0
2 1.8
3
ðm þ 1Þ∑ ðn þ 1Þα mn C n ¼ λ∑ ðn þ 1Þðn þ 2Þα mn Bn þ λðm þ 1Þ∑ ðn þ 1Þα mn C n
Fig. 2 shows the variation of the monosymmetric property parameter along the tapered beam length for three beams with various flange widths. It can be seen from Fig. 2 that for a beam with a wide flange the monosymmetric property parameter exhibits an initial decrease followed by a sharp increase in the value. When buckling occurs, the total energy function reaches to a stationary condition, which requires ∂Π ∂Π ∂Π ¼ ¼ ¼0 ∂An ∂Bn ∂C n
ð19Þ
n¼0
ð23Þ
ql ðm þ n þ 2Þðm þ n þ 3Þðm þ n þ 4Þ
M z βz xmþn Pβk dx ¼ ∑ 2 l ð k þ m þ n þ 1 Þðk þ m þ n þ 2Þ l k¼0 o l
α 4mn ¼ ∫
1.2
qlβk k¼0 ðk þ m þ n þ 1Þðk þ m þ n þ 2Þðk þ m þ n þ 3Þ
ð24Þ
þ∑
1 0.8
where Iyo is the second moment of the cross-sectional area about the y-axis at x = 0, Jo is the torsional constant of the section at x = 0, and βk (k = 0,1, …) are the coefficients of Taylor-series of the monosymmetric property parameter βz. They are defined as follows:
0.6 0.4 0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iyo ¼
x/l Fig. 2. Variation of the monosymmetric parameter with the tapered beam length (tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°).
Jo ¼
t f b3f þ bwo t 3w 12
bf t 3f þ bwo t 3w 3
ð25Þ
ð26Þ
34
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
4. Results and discussion Fig. 4 shows the variation of the lateral–torsional buckling critical loads of web tapered tee-section cantilevers with various lengths. Fig. 4a is for the beam with a tip point load, whereas Fig. 4b is for the beam with a uniformly distributed load. In order to validate the present analytical solution, finite element solutions are also superimposed in Fig. 4. It can be seen from Fig. 4 that, for the beam subject to a tip point load (Fig. 4a) the buckling displacements with the first three terms can provide very good results. The n = 0–2 line agrees with the n = 0–10 line. For the beam subject to a uniformly distributed load (Fig. 4b) the buckling displacements need to include four terms in order to achieve accurate results. Also, it can be observed from Fig. 4b that, for beams shorter than 2 m the present analytical solution does not converge to the finite element solution. The reason for this is probably because the beam is too short so that the traditional bending theory of beams is no longer appropriate. Figs. 5 to 7 compare the critical lateral–torsional buckling loads of tapered tee-section cantilevers with those of un-tapered ones. It is found that the critical buckling loads of the tapered beams can be increased or decreased, depending on the flange width. This pattern happens for both tip point and uniformly distributed load cases and the difference in the buckling loads gradually increases as the length
a
60 n=0-1 n=0-2 n=0-10 FEA
55
Fig. 3. Lateral torsional buckling modes of a tapered tee-section cantilever subjected to (a) a tip point load and (b) a uniformly distributed load (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°, l = 3000 mm, E = 200 GPa, ν = 0.3).
Moment Pl, kN-m
50 45 40 35 30 25
k¼0
xk : l
20 1.5
ð27Þ
Eqs. (19) and (20) are the standard eigen-value equations. For given dimensions of the tapered tee-section beam one can calculate the lowest eigen-value of Eqs. (19) and (20) and thus obtain the critical load of the lateral–torsional buckling of the tapered tee-section beam. 3. Finite element analysis In order to simulate the lateral–torsional buckling behaviour of steel web tapered tee-section cantilevers finite element analyses were carried out using ANSYS software. Tapered tee-section beams with various lengths and flange widths were modelled using four-node shell elements with six degrees of freedom at each node. The boundary conditions of the fixed end support were implemented by forcing all nodes along the web and flange lines to have zero displacements and zero rotations. A concentrated load was applied at the intersection point of the web and flange lines at the free end, whereas a uniformly distributed load was applied on the intersection line of the web and flange. The lateral–torsional buckling modes of a 3 m cantilever with a flange width of 100 mm are presented in Fig. 3a when subjected to a tip point load and in Fig. 3b when subjected to a uniformly distributed load. The results of the finite element analyses are discussed in the following section.
2
2.5
3
3.5
4
4.5
5
5.5
Beam length, m
b
80 n=0-1 n=0-3 n=0-10 FEA
75 70
Moment ql2/2, kN-m
βz ¼ ∑ βk
65 60 55 50 45 40 35 30 1.5
2
2.5
3
3.5
4
4.5
5
5.5
Beam length, m Fig. 4. Comparison of lateral–torsional buckling critical loads obtained by the analytical and the finite element methods for a tapered tee-section cantilever subjected to (a) a tip point load and (b) a uniformly distributed load (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5 o, E = 200 GPa, ν = 0.3).
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
a
a
26
35
50 Tapered tee-section beam Un-tapered tee-section beam
Tapered tee-section beam Un-tapered tee-section beam
24
45
20
Moment Pl, kN-m
Moment Pl, kN-m
22
18 16 14 12
40
35
30
10 25 8 6 1.5
2
2.5
3
3.5
4
4.5
5
5.5
20 1.5
6
2
2.5
3
Beam length, m
b
3.5
4
4.5
5
5.5
6
Beam length, m
b
32
55
Tapered tee-section beam Un-tapered tee-section beam
30
Tapered tee-section beam Un-tapered tee-section beam
28
Moment ql2/2, kN-m
Moment ql2/2, kN-m
50 26 24 22 20 18 16 14
45
40
35
12 10 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Beam length, m Fig. 5. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.19, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
of the beam increases. For example, for a beam with a relatively narrow flange (width/depth = 0.19) the tapering can reduce the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively, as shown in Fig. 5, while for a beam with a relatively wide flange (width/depth = 0.96) the tapering can increase the buckling load by 2%, as shown in Fig. 7. This contradictory result may stem from the fact that, for a cantilever tee-section beam loaded on the flange, tapering brings about two contradictory effects. One is the material reduction, which implies a decrease in the critical load. The other is the reduction of the distance between the centroid and shear centre, which decreases the value of the monosymmetric property parameter and thus implies an increase of the critical load. For the beam with a wide flange the latter effect was found to be predominant; while for the beam with a narrow flange the former effect became dominant. A web tapered tee-section cantilever with an intermediate flange width can increase the buckling loading when the beam is relatively short and decrease it when the beam is relatively long, as shown in Fig. 6. This indicates that not only the flange width but also the beam
30 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Beam length, m Fig. 6. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.38, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
length can affect the influence of the web tapering on the lateral– torsional buckling behaviour of the tee-section cantilevers. Fig. 8 shows the influence of tapering angle on the lateral–torsional buckling loads of a tee-section cantilever whose width-to-depth ratio (bf/(bwo + tf)) is 0.743. As this ratio implies a wide flange, tapering will increase the buckling loads, compared with those of un-tapered sections. The effect of tapering increases as the length of the beam increases and when they are subjected to a tip point load. However the maximum difference shown in Fig. 8 is less than 2% and 4% when subjected to uniformly distributed load and tip point loads respectively. Fig. 9 plots the critical buckling loads of the tapered and un-tapered tee-section cantilevers when subjected to both tip point and uniformly distributed loads. Two features can be found from the figure. The first is that when the critical load curve is plotted using dimensionless parameters, there is almost no difference between the tapered and un-tapered beams. This indicates that the interaction behaviours between the tip point and uniformly distribution loads are the same for both tapered and un-tapered beams. The second is that the critical curves are not the straight lines but slightly arched, which
36
a
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
a
160 150
1.04 α =1o
Tapered tee-section beam Un-tapered tee-section beam
α =2.5o
1.035
α =4o
1.03
qcr,tapered/qcr,untapered
Moment Pl, kN-m
140 130 120 110
1.025 1.02 1.015 1.01
100 1.005 90 1.5
2
2.5
3
3.5
4
4.5
5
5.5
1 1.5
6
2
2.5
Beam length, m
b
b 165
qcr,tapered/qcr,untapered
155
Moment ql2/2, kN-m
3.5
4
4.5
5
4
4.5
5
1.02 α =1o
Tapered tee-section beam Un-tapered tee-section beam
160
3
Beam length, m
150 145 140 135 130
1.018
α =2.5o
1.016
α =4o
1.014 1.012 1.01 1.008 1.006 1.004
125 1.002 120 115 1.5
1 1.5 2
2.5
3
3.5
4
4.5
5
5.5
2
2.5
3
3.5
Beam length, m
6
Beam length, m
implies that the critical load curve can be represented by the equation (P/Pmax) k + (q/qmax) k = 1, where k > 1 is a fitting constant. 5. Conclusions This paper has presented a series of analytical solutions for the lateral–torsional buckling loads of steel web tapered tee-section cantilevers subject to a uniformly distributed load and/or a concentrated load at its free end. The analytical solutions have been validated using the finite element analysis method. From the parametric study the following conclusions have been drawn: • The analytical solutions employed for the lateral displacement and rotation converge rapidly. In practice only a few terms are required to predict the lateral–torsional buckling critical loads of web tapered tee-section cantilevers. • The critical bending moment at the fixed support, when the lateral torsional buckling occurs, gradually decreases with the increase of the beam length. The decrease rate is slightly quicker in the beam with a tip point load than in the beam with a uniformly distributed load.
Fig. 8. Influence of tapering angle on the lateral–torsional buckling loads (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.743, bwo = 350 mm, tw = tf = 10 mm, E = 200 GPa, ν = 0.3).
1 Tapered tee-section beam Un-tapered tee-section beam
0.9
Uniformly distributed load q/qmax
Fig. 7. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.96, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Concentrated load P/Pmax Fig. 9. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to both tip point and uniformly distributed loads (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°, l = 5000 mm, E = 200 GPa, ν = 0.3).
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
• Tapering can increase or decrease the critical buckling loads of web tapered tee-section cantilevers, depending on the flange width. For a beam with a relatively narrow flange (width/depth = 0.19) the tapering can reduce the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively, while for a beam with a relatively wide flange (width/depth = 0.96) the tapering can increase the buckling load by 2%. • The interaction buckling curves between the tip point and uniformly distributed loads are primarily the same for the tapered and un-tapered tee-section cantilevers; both exhibit a slightly convex shape. Acknowledgements The authors would like to thank Doug Wharf, Mike McCulloch and Long-yuan Li for their help and advice on this project. References [1] Bradford MA. Elastic distortional buckling of tee-section cantilevers. Thin-Walled Struct 1999;33:3–17. [2] Anderson JM, Trahair NS. Stability of monosymmetric beams and cantilevers. J Struct Div ASCE 1972;98(ST1):269–86. [3] Nethercot DA. Effective lengths of cantilevers as governed by lateral buckling. Struct Eng 1973;51(5):161–8. [4] Kitipornchai S, Trahair NS. Buckling properties of monosymmetric I-beams. J Struct Div ASCE 1980;106:941–57. [5] Roberts TM, Burt CA. Instability of monosymmetric I-beams and cantilevers. Int J Mech Sci 1985;27(5):313–24.
37
[6] Wang CM, Kitipornchai S. On stability of monosymmetric cantilevers. Eng Struct 1986;8:169–80. [7] Wang CM, Kitipornchai S, Thevendran V. Buckling of braced monosymmetric cantilevers. Int J Mech Sci 1987;29(5):321–37. [8] Trahair NS. Inelastic buckling design of monosymmetric I-beams. Eng Struct 2012;34:564–71. [9] Timoshenko SP. Theory of bending, torsion and buckling of thin-walled members of open cross section. J Franklin Inst 1945;239(5):343–62. [10] Kitipornchai S, Trahair NS. Elastic stability of tapered I-beams. J Struct Div ASCE 1972;98(ST3):713–28. [11] Kitipornchai S, Trahair NS. Elastic behaviour of tapered monosymmetric I-beams. J Struct Div 1975;101(ST8):1661–78. [12] Brown TG. Lateral–torsional buckling of tapered I-beams. J Struct Div ASCE 1981;107(4):689–97. [13] Yang YB, Yau JD. Stability of beams with tapered I-sections. J Eng Mech ASCE 1987;113(9):1337–57. [14] Bradford MA, Cuk PE. Elastic buckling of tapered monosymmetric I-beams. J Struct Eng ASCE 1988;114(5):977–96. [15] Zhang L, Tong GS. Lateral buckling of web-tapered I-beams: a new theory. J Constr Steel Res 2008;64:1379–93. [16] Andrade A, Camotim D. Lateral–torsional buckling of singly symmetric tapered beams: theory and applications. J Eng Mech ASCE 2005;131(6):586–97. [17] Andrade A, Camotim D, Borges Dinis P. Lateral–torsional buckling of singly symmetric web tapered thin-walled I-beams: 1D model vs. shell FEA. Comput Struct 2007;85:1343–59. [18] Raftoyiannis IG, Adamakos T. Critical lateral–torsional buckling moments of steel web-tapered I-beams. Open Constr Build Technol J 2010;4:105–12. [19] Fischer M, Smida M. Coupled instabilities of tapered cantilevers with a t-shaped cross-section. Proceedings of the Third International Conference on Coupled Instabilities in Metal Structures; 2000. p. 379–88. [20] Li LY. Lateral–torsion buckling of cold-formed zed-purlins partial-laterally restrained by metal sheeting. Thin-Walled Struct 2004;42(7):995–1011.
Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
Lateral–torsional buckling of steel web tapered tee-section cantilevers Wei-bin Yuan a, Boksun Kim b,⁎, Chang-yi Chen a a b
College of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou, China School of Marine Science and Engineering, Plymouth University, Plymouth, PL4 8AA, UK
a r t i c l e
i n f o
Article history: Received 17 January 2013 Accepted 29 March 2013 Available online 11 May 2013 Keywords: Lateral–torsional buckling Web tapered tee sections Tapered cantilevers Analytical study Finite element analysis
a b s t r a c t In this paper an analytical model is presented to describe the lateral–torsional buckling behaviour of steel web tapered tee-section cantilevers when subjected to a uniformly distributed load and/or a concentrated load at the free end. To validate the present analytical solutions finite element analyses using ANSYS software are also presented. Good agreement between the analytical and numerical solutions is demonstrated. Using the present analytical solutions, the interactive buckling of the tip point and uniformly distributed loads is investigated and a parametric study is carried out to examine the influence of section dimensions on the critical buckling loads. It is found that web tapering can increase or decrease the critical lateral–torsional buckling loads, depending on the flange width of the beam. For a beam with a wide flange (width/depth = 0.96) the critical buckling load is increased by 2% by web tapering, whereas for a beam with a narrow flange (width/depth = 0.19) web tapering reduces the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Tee-section beams are widely used in modern construction due to their structural efficiency. The main feature of a tee-section beam is the monosymmetry of the cross-section. For most tee-section cantilever beams carrying gravity loading the flange is positioned at the top, in which case the flange is in tension and the unstiffened portion of the web is in compression. Because the neutral axis of a tee-section beam is closer to the flange the maximum compressive stress in the web is much higher than the maximum tensile stress in the flange. This means that such beams fail by compressive stress and the lateral–torsional or lateral–distortional buckling could be one of the main failure modes [1]. The instability of monosymmetric I-beams under various loading conditions has been studied by many researchers [2–8]. The main difficulty of the problem is the presence of an additional torque, owing to the monosymmetry of the section, arising from the pre-buckling longitudinal bending stresses as the beam twists during the buckling. This additional torque causes an effective change in the torsional stiffness of the beam. This feature does not exist in symmetric beams, and was not addressed until the 1940s [9]. Since then different modifications of the torsional stiffness to account for the effect of the additional torque have been proposed [6]. Steel web tapered tee-section beams are very popular because of their aesthetic features and light weight. These beams are mainly cantilevered and have the advantage of low weight-to-strength ratios. They are structurally efficient since the web can be tapered along the beam to closely match the variation of the bending moment of the beam. The depth of ⁎ Corresponding author. Tel.: +44 1752 586135; fax: +44 1752 586101. E-mail address: [email protected] (B. Kim). 0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcsr.2013.03.026
the beam is largest at the fixed support, where its bending moment is greatest, and gradually decreases towards the free end. Although steel web tapered tee-section cantilevers are commonly used, research into the instability of such beams is very limited. The majority of the existing literature deals with the lateral–torsional buckling of tapered I-beams [10–18]. Studies into tapered tee-section cantilevers are few. One rare example is by Fischer and Smida [19]. Kitipornchai and Trahair [10] derived differential equations for the non-uniform torsion of tapered I-beams by analyzing the deformations of the flanges and investigated the elastic flexural–torsional buckling of simply supported tapered I-beams. Later, they extended their method to tapered mono-symmetric I-beams. [11]. Yang and Yau [13], and Bradford and Cuk [14] presented numerical investigations on the lateral–torsional and lateral–distortional buckling of tapered monosymmetric I-beams using finite element methods. Studies by Andrade et al. [16,17] have shown that the lateral torsional buckling loads of simply supported web tapered I-beams were decreased by as much as 20 to 40% as the degree of taper increased. This disagrees with the earlier work by Kitipornchai and Trahair [10], which concluded that the critical loads of web tapered beams did not vary greatly as the degree of taper increased since the torsional stiffness was insensitive to the degree of taper. The disagreement could be because a very short beam of 1.52 m span was used in the work of Kitipornchai and Trahair [10], while Andrade et al. [16,17] used long beams of 6 m, 9 m, and 12 m. The boundary conditions of a beam seem to influence the lateral torsional buckling loads of web tapered I-beams. The buckling loads are decreased for simply supported beams [16–18], while being increased for fix-end beams [18] and cantilevers [16–18], compared with those of un-tapered ones. The increase for the cantilevers is significant and increases as the degree of taper increases. However those three studies
32
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
[16–18] are based on analytical and numerical analyses and hence the results should be validated by experimental work. Web tapered tee-section cantilevers may behave differently from tapered I-beam cantilevers. The lack of a bottom flange means that the lower part of the web is in compression and this increases buckling instability. A study by Fisher and Smida [19] discussed the failure modes of such beams, however their experimental results were not compared to any un-tapered beams. Furthermore no analytical study has been carried out on this topic. In this paper an analytical model is presented to describe the lateral– torsional buckling behaviour of steel web tapered tee-section beams when subjected to a uniformly distributed load and/or a concentrated load at the free end. To validate the present analytical solutions finite element analyses using ANSYS software are also presented. Using the present analytical solutions, the interactive buckling of the distributed and concentrated loads is discussed and a parametric study is carried out to provide the optimum design of tee-section beams against lateral–torsional buckling. 2. Lateral–torsional buckling analysis of steel web tapered tee-section cantilevers Consider a web tapered tee-section cantilever subject to a uniformly distributed load and a concentrated load at its free end, as shown in Fig. 1. Let x be the longitudinal axis of the beam, y and z be the cross-sectional axes parallel to the web and flange, respectively. For convenience, the origin of coordinates was chosen to be the centroid of the section. Due to the tapering of the web, the section properties of the beam are a function of the coordinate x and can be expressed as follows: 2
y¼
Iy ¼
bf t f 2
þ t w ðbwo −x tan α Þ t f þ bwo −x2 tan α bf t f þ t w ðbwo −x tan α Þ
t f b3f ðbwo −x tan α Þt 3w þ 12 12
ð1Þ
and z-axes, respectively, J is the torsional constant of the section, bf is the flange width, tf is the flange thickness, bwo is the web depth at the support (x = 0), tw is the web thickness, and α is the tapering angle. Assume that when lateral–torsional buckling occurs, the displacements of the beam can be described as follows: vðxÞ ¼ ∑ An
xnþ2 xnþ2 l
ð6Þ
xnþ1 l
ð7Þ
wðxÞ ¼ ∑ Bn n¼0
ϕðxÞ ¼ ∑ C n n¼0
where v and w are the transverse and lateral displacements of the beam defined at the shear centre, respectively, ϕ is the angle of rotation of the cross-section, An, Bn and Cn (n = 0, 1, 2, …) are the constants to be determined, and l is the length of the beam. Note that the displacement functions assumed in Eqs. (5)–(7) satisfy the clamped boundary conditions (v = w = ϕ = 0 and dv/dx = dw/dx = 0) at the support (x = 0). The strain energy of a tee-section beam due to the buckling displacements can be calculated using the following formula [20]: l
"
EIz U¼∫ 2 o
d2 v dx2
ð2Þ
" 2 # ðbwo −x tan α Þ2 b −x tan α þ t f −y þ wo 2 12 " 2 2 # tf tf ð3Þ þ y− þ bf t f 12 2
EIy þ 2
!2 # d2 w GJ dϕ 2 þ dx 2 dx dx2
ð8Þ
where E is the Young's modulus and G is the shear modulus. Note that for a tee-section the warping constant is zero and thus no warping energy is involved in Eq. (8). Substituting Eqs. (5)–(7) into Eq. (8) yields l xn 2 xn 2 EI y EI z ∑ An ðn þ 1Þðn þ 2Þ dx þ ∫ 4 ∑ Bn ðn þ 1Þðn þ 2Þ dx 4 l l n¼0 n¼0 o 2l o 2l l xn 2 GJ þ∫ 2 ∑ C n ðn þ 1Þ dx: ð9Þ l 2l n¼0 o l
The loss of the potential energy of the externally applied loads due to the buckling displacements can be calculated using the following formula [2,4,6,20]: l
3
!2
U¼∫
I z ¼ t w ðbwo −x tan α Þ
W ¼ −∫ M z ϕ
3
bf t f þ ðbwo −x tanα Þt w J¼ 3
ð5Þ
l
n¼0
ð4Þ
where y is the distance from the top of the section to the neutral axis, Iy and Iz are the second moments of the cross-sectional area about the y-
o
2 l d2 w 1 dϕ dx− ∫ Mz βz dx 2 2o dx dx
ð10Þ
where Mz is the internal bending moment about the z-axis of the beam in the pre-buckling stage, which is generated due to the externally applied loads, and βz is the parameter describing the monosymmetric property
Fig. 1. A web tapered tee-section cantilever beam subject to a uniformly distributed load and a concentrated load at its free end.
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
33
of the section [2,4,6]. For a beam subject to a uniformly distributed load and a concentrated load at its free end Mz can be expressed as follows:
where Π = U + W is the total potential energy. Substituting Eqs. (9) and (13) into Eq. (14) yields
q 2 Mz ðxÞ ¼ P ðl−xÞ þ ðl−xÞ 2
ðm þ 1Þðm þ 2Þ∑ An ðn þ 1Þðn þ 2Þ∫
ð11Þ
n¼0
where P an q are the concentrated and uniformly distributed loads respectively, shown in Fig. 1. The monosymmetric property parameter is defined as [2,4,6]
βz ¼ ¼
1 Iz
∫ z ydA þ ∫ y dA −2y0
1 Iz
∫ z ydA þ ∫ y dA
A 2
A
! 3
A
EIy xmþn dx 4 l o l
n¼0
l
ðm þ 1Þ∑ C n ðn þ 1Þ∫ n¼0
4 4 tf b t þ w y−t f − bw −y þ t f ¼ y−t f − w þ y− 12I z 2 12I z 2 4I z bf tf 4 4 þ y − y−t f þ 2 y− ð12Þ 4I z 2 bw t 3w
3 t f bf
where bw = bwo–xtanα and y0 is the distance between the shear centre and the centroid, and positive as the convention is upwards, as shown in Fig. 1. Substituting Eqs. (6) and (7) into Eq. (10) yields
ð15Þ
l
n¼0
tf þ 2 y− 2
EIz xmþn dx ¼ 0 l4 l
ðm þ 1Þðm þ 2Þ∑ Bn ðn þ 1Þðn þ 2Þ∫ ¼ ðm þ 1Þðm þ 2Þ∑ C n ∫
3
A
o
l
! 2
l
o
o
Mz xmþnþ1 dx l2 l
ð16Þ
GJ xmþn M xmþnþ1 dx ¼ ∑ Bn ðn þ 1Þðn þ 2Þ∫ 2z dx l l2 l n¼0 o l l
l
þ ðm þ 1Þ∑ C n ðn þ 1Þ∫ n¼0
o
M z β z xmþn dx l l2
ð17Þ
where m = 0, 1, … represents the number of equations. Eqs. (15)–(17) can be further written as follows: Am ¼ 0
ð18Þ 1
ðm þ 1Þðm þ 2Þ∑ ðn þ 1Þðn þ 2Þα mn Bn n¼0
3
¼ λðm þ 1Þðm þ 2Þ∑ α mn C n
xnþ1 xn M ∑ Bn ðn þ 1Þðn þ 2Þ dx W ¼ −∫ 2z ∑ C n l l n¼0 n¼0 o l l
ð13Þ
xn i2 1 M β − ∫ z2 z ∑ C n ðn þ 1Þ dx: 2 o l n¼0 l h
l
2
n¼0
4
ð14Þ
ðn ¼ 0; 1; …Þ
l
1
α mn ¼ ∫ o
¼
α mn ¼ ∫
l
α mn ¼ ∫ o
bf = 100 mm
! 3 1 t w l tan α 1 − ðm þ n þ 1Þ 12Iyo ðm þ n þ 2Þ
GJ xmþn GJ dx ¼ o l l2 l
ð21Þ
! 3 1 t l tan α 1 − w ð22Þ ðm þ n þ 1Þ 3J o ðm þ n þ 2Þ
M z xmþnþ1 P dx ¼ ðm þ n þ 2Þðm þ n þ 3Þ l2 l
þ
bf = 50 mm
1.4
βz(x) / βz(0)
l3 l
2
EI y xmþn dx l4 l
EI yo
o
bf = 250 mm
ð20Þ
where m = 0, 1, … represents the number of equations and λ is the k loading factor and αmn (k = 1, 2, 3, 4) are defined as follows:
3
1.6
n¼0
n¼0
2 1.8
3
ðm þ 1Þ∑ ðn þ 1Þα mn C n ¼ λ∑ ðn þ 1Þðn þ 2Þα mn Bn þ λðm þ 1Þ∑ ðn þ 1Þα mn C n
Fig. 2 shows the variation of the monosymmetric property parameter along the tapered beam length for three beams with various flange widths. It can be seen from Fig. 2 that for a beam with a wide flange the monosymmetric property parameter exhibits an initial decrease followed by a sharp increase in the value. When buckling occurs, the total energy function reaches to a stationary condition, which requires ∂Π ∂Π ∂Π ¼ ¼ ¼0 ∂An ∂Bn ∂C n
ð19Þ
n¼0
ð23Þ
ql ðm þ n þ 2Þðm þ n þ 3Þðm þ n þ 4Þ
M z βz xmþn Pβk dx ¼ ∑ 2 l ð k þ m þ n þ 1 Þðk þ m þ n þ 2Þ l k¼0 o l
α 4mn ¼ ∫
1.2
qlβk k¼0 ðk þ m þ n þ 1Þðk þ m þ n þ 2Þðk þ m þ n þ 3Þ
ð24Þ
þ∑
1 0.8
where Iyo is the second moment of the cross-sectional area about the y-axis at x = 0, Jo is the torsional constant of the section at x = 0, and βk (k = 0,1, …) are the coefficients of Taylor-series of the monosymmetric property parameter βz. They are defined as follows:
0.6 0.4 0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iyo ¼
x/l Fig. 2. Variation of the monosymmetric parameter with the tapered beam length (tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°).
Jo ¼
t f b3f þ bwo t 3w 12
bf t 3f þ bwo t 3w 3
ð25Þ
ð26Þ
34
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
4. Results and discussion Fig. 4 shows the variation of the lateral–torsional buckling critical loads of web tapered tee-section cantilevers with various lengths. Fig. 4a is for the beam with a tip point load, whereas Fig. 4b is for the beam with a uniformly distributed load. In order to validate the present analytical solution, finite element solutions are also superimposed in Fig. 4. It can be seen from Fig. 4 that, for the beam subject to a tip point load (Fig. 4a) the buckling displacements with the first three terms can provide very good results. The n = 0–2 line agrees with the n = 0–10 line. For the beam subject to a uniformly distributed load (Fig. 4b) the buckling displacements need to include four terms in order to achieve accurate results. Also, it can be observed from Fig. 4b that, for beams shorter than 2 m the present analytical solution does not converge to the finite element solution. The reason for this is probably because the beam is too short so that the traditional bending theory of beams is no longer appropriate. Figs. 5 to 7 compare the critical lateral–torsional buckling loads of tapered tee-section cantilevers with those of un-tapered ones. It is found that the critical buckling loads of the tapered beams can be increased or decreased, depending on the flange width. This pattern happens for both tip point and uniformly distributed load cases and the difference in the buckling loads gradually increases as the length
a
60 n=0-1 n=0-2 n=0-10 FEA
55
Fig. 3. Lateral torsional buckling modes of a tapered tee-section cantilever subjected to (a) a tip point load and (b) a uniformly distributed load (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°, l = 3000 mm, E = 200 GPa, ν = 0.3).
Moment Pl, kN-m
50 45 40 35 30 25
k¼0
xk : l
20 1.5
ð27Þ
Eqs. (19) and (20) are the standard eigen-value equations. For given dimensions of the tapered tee-section beam one can calculate the lowest eigen-value of Eqs. (19) and (20) and thus obtain the critical load of the lateral–torsional buckling of the tapered tee-section beam. 3. Finite element analysis In order to simulate the lateral–torsional buckling behaviour of steel web tapered tee-section cantilevers finite element analyses were carried out using ANSYS software. Tapered tee-section beams with various lengths and flange widths were modelled using four-node shell elements with six degrees of freedom at each node. The boundary conditions of the fixed end support were implemented by forcing all nodes along the web and flange lines to have zero displacements and zero rotations. A concentrated load was applied at the intersection point of the web and flange lines at the free end, whereas a uniformly distributed load was applied on the intersection line of the web and flange. The lateral–torsional buckling modes of a 3 m cantilever with a flange width of 100 mm are presented in Fig. 3a when subjected to a tip point load and in Fig. 3b when subjected to a uniformly distributed load. The results of the finite element analyses are discussed in the following section.
2
2.5
3
3.5
4
4.5
5
5.5
Beam length, m
b
80 n=0-1 n=0-3 n=0-10 FEA
75 70
Moment ql2/2, kN-m
βz ¼ ∑ βk
65 60 55 50 45 40 35 30 1.5
2
2.5
3
3.5
4
4.5
5
5.5
Beam length, m Fig. 4. Comparison of lateral–torsional buckling critical loads obtained by the analytical and the finite element methods for a tapered tee-section cantilever subjected to (a) a tip point load and (b) a uniformly distributed load (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5 o, E = 200 GPa, ν = 0.3).
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
a
a
26
35
50 Tapered tee-section beam Un-tapered tee-section beam
Tapered tee-section beam Un-tapered tee-section beam
24
45
20
Moment Pl, kN-m
Moment Pl, kN-m
22
18 16 14 12
40
35
30
10 25 8 6 1.5
2
2.5
3
3.5
4
4.5
5
5.5
20 1.5
6
2
2.5
3
Beam length, m
b
3.5
4
4.5
5
5.5
6
Beam length, m
b
32
55
Tapered tee-section beam Un-tapered tee-section beam
30
Tapered tee-section beam Un-tapered tee-section beam
28
Moment ql2/2, kN-m
Moment ql2/2, kN-m
50 26 24 22 20 18 16 14
45
40
35
12 10 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Beam length, m Fig. 5. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.19, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
of the beam increases. For example, for a beam with a relatively narrow flange (width/depth = 0.19) the tapering can reduce the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively, as shown in Fig. 5, while for a beam with a relatively wide flange (width/depth = 0.96) the tapering can increase the buckling load by 2%, as shown in Fig. 7. This contradictory result may stem from the fact that, for a cantilever tee-section beam loaded on the flange, tapering brings about two contradictory effects. One is the material reduction, which implies a decrease in the critical load. The other is the reduction of the distance between the centroid and shear centre, which decreases the value of the monosymmetric property parameter and thus implies an increase of the critical load. For the beam with a wide flange the latter effect was found to be predominant; while for the beam with a narrow flange the former effect became dominant. A web tapered tee-section cantilever with an intermediate flange width can increase the buckling loading when the beam is relatively short and decrease it when the beam is relatively long, as shown in Fig. 6. This indicates that not only the flange width but also the beam
30 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Beam length, m Fig. 6. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.38, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
length can affect the influence of the web tapering on the lateral– torsional buckling behaviour of the tee-section cantilevers. Fig. 8 shows the influence of tapering angle on the lateral–torsional buckling loads of a tee-section cantilever whose width-to-depth ratio (bf/(bwo + tf)) is 0.743. As this ratio implies a wide flange, tapering will increase the buckling loads, compared with those of un-tapered sections. The effect of tapering increases as the length of the beam increases and when they are subjected to a tip point load. However the maximum difference shown in Fig. 8 is less than 2% and 4% when subjected to uniformly distributed load and tip point loads respectively. Fig. 9 plots the critical buckling loads of the tapered and un-tapered tee-section cantilevers when subjected to both tip point and uniformly distributed loads. Two features can be found from the figure. The first is that when the critical load curve is plotted using dimensionless parameters, there is almost no difference between the tapered and un-tapered beams. This indicates that the interaction behaviours between the tip point and uniformly distribution loads are the same for both tapered and un-tapered beams. The second is that the critical curves are not the straight lines but slightly arched, which
36
a
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
a
160 150
1.04 α =1o
Tapered tee-section beam Un-tapered tee-section beam
α =2.5o
1.035
α =4o
1.03
qcr,tapered/qcr,untapered
Moment Pl, kN-m
140 130 120 110
1.025 1.02 1.015 1.01
100 1.005 90 1.5
2
2.5
3
3.5
4
4.5
5
5.5
1 1.5
6
2
2.5
Beam length, m
b
b 165
qcr,tapered/qcr,untapered
155
Moment ql2/2, kN-m
3.5
4
4.5
5
4
4.5
5
1.02 α =1o
Tapered tee-section beam Un-tapered tee-section beam
160
3
Beam length, m
150 145 140 135 130
1.018
α =2.5o
1.016
α =4o
1.014 1.012 1.01 1.008 1.006 1.004
125 1.002 120 115 1.5
1 1.5 2
2.5
3
3.5
4
4.5
5
5.5
2
2.5
3
3.5
Beam length, m
6
Beam length, m
implies that the critical load curve can be represented by the equation (P/Pmax) k + (q/qmax) k = 1, where k > 1 is a fitting constant. 5. Conclusions This paper has presented a series of analytical solutions for the lateral–torsional buckling loads of steel web tapered tee-section cantilevers subject to a uniformly distributed load and/or a concentrated load at its free end. The analytical solutions have been validated using the finite element analysis method. From the parametric study the following conclusions have been drawn: • The analytical solutions employed for the lateral displacement and rotation converge rapidly. In practice only a few terms are required to predict the lateral–torsional buckling critical loads of web tapered tee-section cantilevers. • The critical bending moment at the fixed support, when the lateral torsional buckling occurs, gradually decreases with the increase of the beam length. The decrease rate is slightly quicker in the beam with a tip point load than in the beam with a uniformly distributed load.
Fig. 8. Influence of tapering angle on the lateral–torsional buckling loads (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.743, bwo = 350 mm, tw = tf = 10 mm, E = 200 GPa, ν = 0.3).
1 Tapered tee-section beam Un-tapered tee-section beam
0.9
Uniformly distributed load q/qmax
Fig. 7. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to (a) a tip point load and (b) a uniformly distributed load (bf/(bwo + tf) = 0.96, bwo = 250 mm, tw = tf = 10 mm, α = 2.5°, E = 200 GPa, ν = 0.3).
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Concentrated load P/Pmax Fig. 9. Comparison of lateral–torsional buckling loads between tapered and un-tapered tee-section cantilevers subjected to both tip point and uniformly distributed loads (bf = 100 mm, tf = 10 mm, bwo = 250 mm, tw = 10 mm, α = 2.5°, l = 5000 mm, E = 200 GPa, ν = 0.3).
W. Yuan et al. / Journal of Constructional Steel Research 87 (2013) 31–37
• Tapering can increase or decrease the critical buckling loads of web tapered tee-section cantilevers, depending on the flange width. For a beam with a relatively narrow flange (width/depth = 0.19) the tapering can reduce the buckling load up to10% and 6% for the tip point loading and the uniformly distributed load respectively, while for a beam with a relatively wide flange (width/depth = 0.96) the tapering can increase the buckling load by 2%. • The interaction buckling curves between the tip point and uniformly distributed loads are primarily the same for the tapered and un-tapered tee-section cantilevers; both exhibit a slightly convex shape. Acknowledgements The authors would like to thank Doug Wharf, Mike McCulloch and Long-yuan Li for their help and advice on this project. References [1] Bradford MA. Elastic distortional buckling of tee-section cantilevers. Thin-Walled Struct 1999;33:3–17. [2] Anderson JM, Trahair NS. Stability of monosymmetric beams and cantilevers. J Struct Div ASCE 1972;98(ST1):269–86. [3] Nethercot DA. Effective lengths of cantilevers as governed by lateral buckling. Struct Eng 1973;51(5):161–8. [4] Kitipornchai S, Trahair NS. Buckling properties of monosymmetric I-beams. J Struct Div ASCE 1980;106:941–57. [5] Roberts TM, Burt CA. Instability of monosymmetric I-beams and cantilevers. Int J Mech Sci 1985;27(5):313–24.
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[6] Wang CM, Kitipornchai S. On stability of monosymmetric cantilevers. Eng Struct 1986;8:169–80. [7] Wang CM, Kitipornchai S, Thevendran V. Buckling of braced monosymmetric cantilevers. Int J Mech Sci 1987;29(5):321–37. [8] Trahair NS. Inelastic buckling design of monosymmetric I-beams. Eng Struct 2012;34:564–71. [9] Timoshenko SP. Theory of bending, torsion and buckling of thin-walled members of open cross section. J Franklin Inst 1945;239(5):343–62. [10] Kitipornchai S, Trahair NS. Elastic stability of tapered I-beams. J Struct Div ASCE 1972;98(ST3):713–28. [11] Kitipornchai S, Trahair NS. Elastic behaviour of tapered monosymmetric I-beams. J Struct Div 1975;101(ST8):1661–78. [12] Brown TG. Lateral–torsional buckling of tapered I-beams. J Struct Div ASCE 1981;107(4):689–97. [13] Yang YB, Yau JD. Stability of beams with tapered I-sections. J Eng Mech ASCE 1987;113(9):1337–57. [14] Bradford MA, Cuk PE. Elastic buckling of tapered monosymmetric I-beams. J Struct Eng ASCE 1988;114(5):977–96. [15] Zhang L, Tong GS. Lateral buckling of web-tapered I-beams: a new theory. J Constr Steel Res 2008;64:1379–93. [16] Andrade A, Camotim D. Lateral–torsional buckling of singly symmetric tapered beams: theory and applications. J Eng Mech ASCE 2005;131(6):586–97. [17] Andrade A, Camotim D, Borges Dinis P. Lateral–torsional buckling of singly symmetric web tapered thin-walled I-beams: 1D model vs. shell FEA. Comput Struct 2007;85:1343–59. [18] Raftoyiannis IG, Adamakos T. Critical lateral–torsional buckling moments of steel web-tapered I-beams. Open Constr Build Technol J 2010;4:105–12. [19] Fischer M, Smida M. Coupled instabilities of tapered cantilevers with a t-shaped cross-section. Proceedings of the Third International Conference on Coupled Instabilities in Metal Structures; 2000. p. 379–88. [20] Li LY. Lateral–torsion buckling of cold-formed zed-purlins partial-laterally restrained by metal sheeting. Thin-Walled Struct 2004;42(7):995–1011.