Buckling of composite columns of lipped-channel and hat sections with web stiffener

ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 1149–1160

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Buckling of composite columns of lipped-channel and hat sections with web stiffener Huu Thanh Nguyen, Seung Eock Kim  Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea

a r t i c l e in fo

abstract

Article history: Received 23 July 2008 Received in revised form 25 April 2009 Accepted 28 April 2009 Available online 23 May 2009

The buckling of thin-walled composite columns in hat sections and lipped-channel sections reinforced with web stiffener is studied. The columns were composed of symmetric angle-ply laminates. The finite element method was used to investigate the buckling behaviour of the columns. Bifurcation analyses were carried out to obtain the buckling load and mode shapes of the columns. Load-deflection analyses were performed to study the post-buckling behaviour of the columns. The results showed significant effects of ply angle and geometric parameters on the buckling and post-buckling behaviour of the columns. This research provides a guide for improving the loading capacity of composite columns. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Composite column Web stiffener Ply angle Buckling Finite element method

1. Introduction As the use of composite structures made of fiber-reinforced plastic (FRP) increases, there has been more research on composite structural products. Several studies have investigated the behaviour of composite sections under compression. Turvey and Wittrick [1] investigated the effect of orthotropy on the stability of structures composed of plate elements. Lee [2] studied the local buckling of orthotropic rectangular tubes, channels, Z- and I-sections with exact and approximate analyses. The effects of local and overall imperfections were considered by Upadhya and Loughlan [3]. They investigated the interactive buckling of orthotropic channel sections. Bauld and Tzeng [4] extended Vlasov’s thin-walled beam theory to analyze fiber-reinforced members with open cross-sections. Reddy et al. [5] studied the design of composite I-sections under axial compression based on the correlation of NASTRAN analysis with experimental data. Bonanni et al. [6] tested mostly quasi-isotropic channels, Z-, I- and J-sections and modeled specimens with a shell-analysis code, investigating crippling modes. Raftoyiannis [7] and Godoy et al. [8] studied the interaction of buckling modes in composite columns, showing significant effects on the structural behaviour and loading capacity of thin-walled composite members related to the deformation of cross-sections. Bank et al. [9] performed an experimental investigation of local compression flange buckling

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E-mail address: [email protected] (S.E. Kim). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.04.011

and the failure of commercially produced pultruded fiberreinforced plastic I-shaped beams. An experimental and theoretical study on post-buckling and crippling of composite I-sections was presented by Parnas et al. [10]. In their study, the postbuckling stiffness was calculated based on the load-carrying capacity of the web. Barbero et al. [11] employed the finite element method to investigate post-buckling of plate, I-section, angle section and a unicolumn (a square box beam with flanges on the outside). Barbero and Trovillion [12] modeled pultruded E-glass I-sections with the finite element method and tested specimens. Ishikawa et al. [13] conducted finite element analyses and experiments for predicting linear buckling and post-buckling behaviour of carbon fiber-reinforced plastic T stiffeners. Barbero [14] and Barbero et al. [15] investigated, both analytically and experimentally, the effect of local deformation on the buckling behaviour of thin-walled composite members. In these studies, the authors employed the finite element method, and their analyses were performed using the ABAQUS code. Their investigations showed that cross-section deformation may significantly influence the buckling behaviour of members. Hilburger and Starnes [16] investigated the effect of imperfections on the buckling response of compression-loaded composite shells. Their study indicated that the laminate stacking sequence affected buckling and post-buckling behaviour of composite members. Recently, Silvestre and Camotim [17,18] developed the first- and second-order generalized beam theory (GBT) for arbitrary orthotropic materials. They presented formulations for the analyses of bucking behaviour for thin-walled composite members made of laminate plates displaying arbitrary orthotropy. These authors

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[19] used GBT to investigate the buckling behaviour of thin-walled composite columns. Silva and Silvestre [20] applied GBT to analyze the linear and buckling behaviour of FRP composite thin-walled members. They studied the influence of material couplings on the linear and buckling behaviour of I-section composite columns. Their work provided a detailed discussion of the mechanical roots on the coupling effects between GBT deformation modes participating in buckling modes. Teter and Kolakowski [21] conducted research dealing with the buckling of prismatic thin-walled composite columns in open cross-sections with web stiffener. They developed an analytical method relying on Koiter’s asymptotic theory. The stiffness of the thin-walled composite member was derived by employing classical composite laminate and plate theories. The governing equations of thinwalled composite members were established by applying the principal of virtual work and solved by the asymptotic Byskov– Hutchinson method. This approach for non-linear approximation can evaluate the effect of imperfection and interaction on buckling. This paper investigates the buckling and post-buckling behaviour of thin-walled composite columns subjected to compressive loading. These columns consist of hat sections and lipped-channel sections with web stiffener. The buckling loads and corresponding mode shapes of the columns with different geometric parameters and ply angles were investigated through bifurcation analysis. Load-deflection analysis was also carried out to study the postbuckling behaviour and the effect of local buckling on the loading capacity of the columns. In this study, finite element analyses were performed by using the commercial finite element package ABAQUS [22].

Fig. 2. Composite columns of lipped-channel section: (a) without web stiffener and (b) with web stiffener.

2. Geometry and the finite element model of composite columns The composite columns of two types of cross-section shapes, hat section (Fig. 1) and lipped-channel section (Fig. 2) were

Fig. 1. Composite columns of hat section: (a) without web stiffener and (b) with web stiffener.

studied. Teter and Kolakowski [21]’s column dimensions were employed. The sectional dimensions are shown in Figs. 3 and 4, where the width (b) and the height (h) of the sections are 50 and 25 mm, respectively. The thickness (t) is 1 mm. Non-dimensional parameters a ¼ bs/b, b ¼ b1/b, g ¼ hs/h and l ¼ L/r are defined, where bs is the width of the web stiffener, b1 the width of the lip, hs the height of the stiffener and r minimum radius of gyration of section. a ¼ 0 (g ¼ 0) indicates un-stiffened section and aa0 (ga0) means stiffened section. In general, there are two ways to analyze the laminated section. The first explicitly defines thickness, material and fiber orientation for each layer. The other uses equivalent properties for the whole layer. Throughout this research, the former method was employed to obtain more accurate results. Laminate of 1 mm thickness that was composed of eight layers of 0.125 mm thickness was used in this work. Each layer was made of Glass–Epoxy composite material with mechanical properties of E1 ¼ 140 GPa, E2 ¼ 10.3 GPa, G12 ¼ G13 ¼ 5.15 GPa, G23 ¼ 4.63 GPa and n12 ¼ 0.29. The symmetric angle-ply [(y,y)2]s was used, where the ply angle (y) is defined as the angle between the fiber and the column axis as shown in Fig. 5(a). The lamina stacking sequence is shown in Fig. 5(b). The composite column was modeled by the shell element in ABAQUS [22]. The finite element model of the columns is shown in Fig. 6. The element size was 8 mm. Initially, the shell element S4R was used to model the columns. It worked well for eigenvalue analysis but there was a convergence problem with geometric non-linear analysis. Therefore, the shell element S8R was used instead. The element S8R is eight nodes shear deformable shell element with quadratic shape functions. Each node of the element has six degrees of freedom. This element allows large displacements but small strains. Therefore it is more appropriate for geometry non-linear analysis.

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1151

Fig. 3. Hat section: (a) without web stiffener (a ¼ 0, g ¼ 0) and (b) with web stiffener (aa0, ga0).

Fig. 4. Lipped-channel section: (a) without web stiffener (a ¼ 0, g ¼ 0) and (b) with web stiffener (aa0, ga0).

Fig. 5. (a) Ply angle (y) and (b) symmetric angle-ply laminate [(y,y)2]s.

Rigid body constrains [22] were applied to the nodes on the end sections of the columns as illustrated in Fig. 6. The displacements of the sections were represented by reference nodes located at centroid of section. With these constrains the end sections were planar during the analysis since the ends of the column were stiffened by diaphragms in design practice. The right angle between component walls and end section plane remained. Boundary conditions were applied to the reference nodes. As shown in Fig. 6, three displacements and axial rotation of the bottom reference node and two lateral displacements of the top reference node were fixed. The other degrees of freedom were free. The compressive force was applied to the reference node at the top of the column.

3. Eigenvalue analysis In this section, we conduct eigenvalue analysis to obtain the buckling load and corresponding mode shapes of the columns in Figs. 1 and 2. The width of the lip and the height of the stiffener

are b1 ¼ 12.5 mm and hs ¼ 4 mm, and their corresponding parameters are b ¼ 0.25 and g ¼ 0.16, respectively. The widths of the stiffener are bs ¼ 0, 4, 8, 12 mm and their stiffener corresponding parameters are a ¼ 0, 0.08, 0.16, 0.24, respectively. The column length is L ¼ 650 mm. The nominal critical stress (scr) and the number of buckling halfwaves (n) formed along the column are presented with respect to the ply angle (y) and stiffener parameter (a). The number of buckling halfwaves (n) is counted from the buckling mode shape. Fig. 7 shows the global buckling mode shapes of the composite columns. The mode shapes A, B, C are associated with the columns of the hat section while the mode shapes D, E, F are of the lippedchannel section. Mode shapes A and D are flexural–torsional modes, B and E are distortional modes, and C and F are flexural modes. The global buckling stresses (scr) of the hat-section and lipped-channel-section columns with respect to the ply angle (y) are shown in Figs. 8 and 9, respectively. The flexural–torsional buckling mode showed lower critical stress than the flexural mode for the hat-section column and vice versa for the lippedchannel-section column. When the ply angle (y) is 451 the flexural buckling mode changes to the distortional mode for both the hatsection and lipped-channel-section columns. The columns with different stiffener width showed similar buckling stress in flexural buckling mode. The local buckling mode shapes of the hat-section and lippedchannel-section columns with respect to the ply angle (y) and stiffener parameter (a) are shown in Figs. 10 and 11, respectively. Local buckling occurs in both the web and the flange for the unstiffened sections (Figs. 10(a,b,c) and 11(a,b,c)), while it only occurs in the flange for the stiffened sections (Figs. 10(d,e,f) and 11(d,e,f)). The results show that the local buckling stresses (scr) and number of halfwaves (n) of the hat-section and lippedchannel-section columns are almost identical. The local buckling stresses (scr) and the number of halfwaves (n) with respect to the ply angle (y) and stiffener parameter (a) are shown in Figs. 12

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Reference point located at centroid of section

Two lateral displacements are restrained

Top end

Three Reference displacements and point located at axial rotation are centroid of restrained section Bottom end

Two lateral displacements are restrained

Reference point located at centroid of section

Top end

Reference point located at centroid of section

Three displacements and axial rotation are restrained

Bottom end

Fig. 6. Finite element model and boundary conditions: (a) column of hat section and (b) column of lipped-channel section.

Mode shape A

Mode shape D

Mode shape B

Mode shape E

Mode shape C

Mode shape F

Fig. 7. Global buckling mode shapes of the composite columns under axial compression. (a) Flexural–torsional mode, (b) Distorsional mode, (c) Flexural mode.

and 13, respectively. In Fig. 12, the buckling stress curves showed the lowest critical stress at y ¼ 901 and the highest at y(max) ¼ 43.91. The critical stresses (scr) for a ¼ 0.08, 0.16, 0.24 were close to one another, and they were 2–2.5 times higher than at a ¼ 0. The number of halfwaves (n) increased as the ply angle (y) increased as shown in Fig. 13. The number of halfwaves (n) for

stiffened sections was higher than un-stiffened sections. The stiffened sections had an identical number of halfwaves regardless the stiffener size. The lowest and highest number of halfwaves (n) occurred at the ply angle of 01 and 901, respectively. Next, eigenvalue analysis was performed on the composite columns with the sections in Figs. 3 and 4 for different stiffener

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160

350

α=0 α = 0.08 α = 0.16 α = 0.24

140 120 100

250

Flexural-torsional mode Shape A

80

α=0 α = 0.08 α = 0.16 α = 0.24

300

σcr [Mpa]

σcr [Mpa]

1153

200

Flexural-torsional mode Shape D

150

60 100

40

Lipped-channel section

50

Hat section

20

0

0

0

0

10

20

30

40

50

60

70

80

10

20

30

90

θ [degree]

40 50 θ [degree]

60

70

80

90

180

250 Distortional mode Shape B

α = 0.08 α = 0.16

120

150 Flexural mode Shape C

100

α=0

140

σcr [Mpa]

σcr [Mpa]

200

Distortional mode Shape E

160

α=0 α = 0.08 α = 0.16 α = 0.24

α = 0.24

100 80

Flexural mode Shape F

60 40 Lipped-channel section

50 20

Hat section

0

0 0

10

20

30

40

50

60

70

80

90

θ [degree] Fig. 8. Global buckling stress scr (MPa) with respect to ply angle (y) and stiffener parameter (a) for the composite columns of the hat section under axial compression. (a) Flexural–torsional mode. (b) Flexural and distortional modes.

parameter a. In this analysis, the parameter a varied from 0 to 0.8 while the other parameters were taken as b ¼ 0.25 and g ¼ 0.16. The buckling stress associated with the first buckling mode of the hat-section column is shown with respect to the ply angle (y) and stiffener parameter (a) in Fig. 14. The buckling stress of the unstiffened section (a ¼ 0) was governed by local buckling at 01ryr22.81, flexural–torsional buckling (shape A) at 22.81ryr901. The maximum buckling stress was obtained at y ¼ 22.81. The buckling stress of the stiffened section with 0oar0.64 was governed by local buckling at 01ryr101, flexural–torsional buckling in 101ryr901. The maximum buckling stress was y ¼ 101. For a40.64 local buckling occurred in the web of the stiffener and the maximum buckling stress was obtained at 101ryr281. The buckling stress of the lippedchannel-section column with respect to the ply angle (y) and the stiffener parameter (a) is shown in Fig. 15. The buckling stress of the un-stiffened section (a ¼ 0) was governed by local buckling at 01ryr301, distortional buckling at 301ryr451 and flexural buckling at 451ryr901. The maximum buckling stress was obtained at y ¼ 301. The buckling stress of the stiffened section with 0oar0.5 was governed by local buckling at 01ryr121, distortional buckling at 121ryr451 and flexural buckling at 451ryr901. The maximum buckling stress is obtained at y ¼ 121. Local buckling occurred in the web of the stiffener for a40.5. In

0

10

20

30

40 50 θ [degree]

60

70

80

90

Fig. 9. Global buckling stress scr (MPa) with respect to ply angle (y) and stiffener parameter (a) for the composite columns of the lipped-channel section under axial compression. (a) Flexural–torsional mode. (b) Flexural and distortional modes.

this case, the maximum buckling stress was obtained at 101ryo301. The maximum buckling stress of the hat-section and lipped-channel-section columns with respect to the stiffener parameter (a) is presented in Table 1. The maximum buckling stress was obtained at the transition angles of the mode shapes. The maximum buckling stress of the stiffened section was 64% and 39% higher than the un-stiffened section for the hat section and lipped-channel section, respectively. The minimum buckling stress was at y ¼ 901 for all cases. In Figs. 14 and 15, the stiffener width had a significant effect on the local and flexural–torsional buckling modes of the hat section for yr451. It did not have a significant influence on the global distortional and flexural modes of lipped-channel section. The buckling of the columns was also investigated for the effect of the lip width (b1) of the section. The parameters of a ¼ 0.08, g ¼ 0.16 and 0rbr0.3 were analyzed. The buckling stress with respect to the ply angle (y) and the lip parameter (b) is shown in Figs. 16 and 17 for the hat section and lipped channel section, respectively. The width of the lip had a significant effect on the critical stress in the range yr351, where the local buckling or distortional buckling was governing mode. In the range y4351, the lip width had no significant effect on the critical stress associated with the flexural–torsional buckling mode of the hat section and the flexural buckling mode of the lipped-channel

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α = 0, θ = 0°

α = 0, θ = 45°

α = 0, θ = 90°

α ≠ 0, θ = 0°

α ≠ 0, θ = 45°

α ≠ 0, θ = 90°

Fig. 10. Local buckling mode shapes of the composite columns of the hat section under axial compression (un-stiffened sections: (a)(b)(c), stiffened sections: (d)(e)(f)).

α = 0, θ = 0°

α ≠ 0, θ = 0°

α = 0, θ = 45°

α ≠ 0,θ = 45°

α = 0, θ = 90°

α ≠ 0, θ = 90°

Fig. 11. Local buckling mode shapes of the composite columns for the lipped-channel section under axial compression with respect to ply angles (y) (un-stiffened sections: (a)(b)(c), stiffened sections: (d)(e)(f)).

section. The maximum buckling stress was reached at b ¼ 0.2, y ¼ 51 and b ¼ 0.25, y ¼ 121 for the hat section and lippedchannel section, respectively. The maximum buckling stress of the hat section and lipped-channel section with respect to parameter

b is presented in Table 2. Making a comparison between the highest buckling stress and the lowest in Table 2, the lip of the section can increase the buckling stress up to 3 and 3.2 times for the hat section and the lipped-channel section, respectively.

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300 250

Local buckling mode

140

Distortional buckling mode Shape E

120

α=0 α = 0.04 α = 0.08 α = 0.16 α = 0.24 α = 0.32 α = 0.4 α = 0.5 α = 0.56 α = 0.64 α = 0.72 α = 0.8

Lipped-channel section

100

150

σcr [MPa]

σcr [Mpa]

200

12°

160

α=0 α = 0.08 α = 0.16 α = 0.24

1155

100 50 0 10

20

30

40

50

60

70

60

80

90

θ [degree]

b1 = 12.5; β = 0.25 hs = 4; γ = 0.16 L = 650; λ = 36.1

Local buckling mode

40

θ (max) = 43.9° 0

Flexural buckling mode Shape F

80

20 0

Fig. 12. Local buckling stress scr (MPa) with respect to ply angle (y) and stiffener parameter (a) for both the hat-section and lipped-channel-section columns.

0

10

20

30

40 50 θ [degree]

60

70

80

90

Fig. 15. Buckling stress scr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the lipped-channel section.

45 α=0 α = 0.08 α = 0.16 α = 0.24

40

Number of halfwave (n)

35 30 25 20 15 10 5 0 10

0

20

30

40 50 θ [degree]

60

70

80

90

Fig. 13. Number of halfwaves (n) with respect to ply angle (y) and stiffener parameter (a) for both the hat-section and lipped-channel-section columns.

10° 160 140

Local buckling mode

α=0 α = 0.04 α = 0.08 α = 0.16 α = 0.24 α = 0.32 α= 0.4 α = 0.5 α = 0.56 α = 0.64 α = 0.72 α = 0.8

Hat section

120 σcr [MPa]

100 Flexural-torsional buckling mode Shape A

80 60 40

b1 = 12.5; β = 0.25 hs = 4; γ = 0.16 L = 650; λ = 36.1

Local buckling mode

20

22.8°

0 0

10

20

30

40 50 θ [degree]

60

70

80

90

Fig. 14. Buckling stress scr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the hat section.

Eigenvalue analysis was performed on the composite columns in Figs. 1 and 2 to investigate the effect of the stiffener height (hs). The sections in Figs. 3 and 4 for various hs were analyzed. In this analysis, the value of a ¼ 0.08 and b ¼ 0.25 was used. The parameter g varied from 0 to 1. Figs. 18 and 19 show the buckling stress with respect to the ply angle (y) and parameter (g) for the hat section and lipped-channel section, respectively. The maximum buckling stress with respect to parameter g is presented in Table 3. The results show that the height of the stiffener can increase significantly the buckling stress of the columns. The critical stress increased slowly at 0.08rgr0.5 but rose quickly at g40.5. The stiffener height was able to increase the maximum critical stress of up to 1.7 and 2.2 times for the hat section and lipped-channel section, respectively. For 0.08rgr1, the maximum critical stress was obtained at y ¼ 101 and 121ryr17.51 for the hat section and lipped-channel section, respectively. The height of the stiffener had a greater effect on the lipped-channel section in comparison with the hat section. Thus, an increase in the height of the stiffener leads to a larger increase in stiffness for flexural buckling than torsional buckling. The variation of the buckling stress of the columns with respect to the column length (L) was investigated. In this investigation, the sectional parameters were taken at a ¼ 0, 0.24, b ¼ 0.25 and g ¼ 0.16. The slenderness ratio (l) varied from 5.6 to 222.3 corresponding to L varied from 100 to 4000 mm. Figs. 20 and 21 showed the buckling stress with respect to the ply angle (y) and slenderness ratio (l) for the un-stiffened and the stiffened hat section, respectively. The curves included two parts. The first part was associated with the local buckling and the other part the global flexural–torsional buckling. The local buckling parts of the curves with yr201 had a rather long range of 0olo40–55, whereas, the other curves had short ones, 0olo12. The critical stress had a high value for the local buckling range but it dropped dramatically after entering the global buckling range. The highest critical stress was obtained at y ¼ 451. The critical stress of the stiffened section was greater than that of the un-stiffened section from 1.8 to 2.6 times. Figs. 22 and 23 show the buckling stress with respect to the ply angle (y) and slenderness ratio (l) for the lipped-channel section. Similar results with the hat section were obtained for the lipped-channel section. However, in comparison with the hat section the local buckling range of the lipped-channel section was longer by 1.5 and 1.25 times for un-stiffened and stiffened section, respectively.

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Table 1 Maximum buckling stress scr (MPa) vs. stiffener parameter a. Hat-section

a scr (max) MPa

0. 84.7 22.81

y (max)

0.04 118.5 101

0.08 120.8 101

0.16 125.5 101

0.24 129.0 101

0.32 131.4 101

0.4 133.8 101

0.5 136.9 101

0.56 138.8 101

0.64 141.3 101

0.72 133.4 151

0.8 112.8 201

0.04 132.4 121

0.08 133.9 121

0.16 136.3 121

0.24 138.3 121

0.32 138.1 121

0.4 137.8 121

0.5 136.6 121

0.56 138.0 101

0.64 135.6 101

0.72 127.9 151

0.8 116.4 201

Lipped-channel section

a scr (max) MPa

0. 98.8 301

y (max) b ¼ 0.25, g ¼ 0.16.

10°

160

140 β=0 β = 0.05 β = 0.1 β = 0.15 β = 0.2 β = 0.25 β = 0.3

Hat section

σcr [MPa]

100 80 60

Flexural-torsional buckling mode

Local or distortional buckling mode

40

140

bs = 4; α = 0.08 hs = 4; γ = 0.16 L = 650; λ = 3 6.1

100 80

Flexural-torsional buckling mode

60 40

20 35° 0

10

20

30

22.8°

0

40 50 θ [degree]

60

70

80

0

90

Fig. 16. Buckling stress scr (Mpa) with respect to ply angle (y) and parameter (b) for the hat section.

10

250 Local buckling mode

β=0 β = 0.05 β = 0.1

Lipped-channel section

β = 0.15

σcr [Mpa]

β = 0.3

60

Flexural buckling mode

Local or distortional buckling mode

40

bs = 4; α = 0.08 hs = 4; γ = 0.16 L = 650; λ = 36.1

35° 10

20

30

80

90

γ=0 γ = 0.08 γ = 0.16 γ = 0.24 γ = 0.32 γ = 0.4 γ = 0.5 γ = 0.6 γ = 0.8 γ=1

bs = 4; α = 0.08 b1 = 12.5; β = 0.25 L = 650; λ = 36.1

100

50

0

70

150

Flexural buckling mode

Local buckling mode

20 0

60

Lipped-channel section

β = 0.2

80

40 50 θ [degree]

Distortional buckling mode

200

β = 0.25

σcr [Mpa]

30

12° 17.5°

12°

120

20

Fig. 18. Buckling stress scr (Mpa) with respect to ply angle (y) and parameter (g) for the hat section.

140

100

bs = 4; α = 0.08 b1 = 12.5; β = 0.25 L = 650; λ = 36.1

Local buckling mode

20

0

γ=0 γ = 0.08 γ = 0.16 γ = 0.24 γ = 0.32 γ = 0.4 γ = 0.5 γ = 0.6 γ = 0.8 γ=1

Hat section

120

σcr [MPa]

120

10° Local buckling mode

0

40 50 θ [degree]

60

70

80

90

Fig. 17. Buckling stress scr (Mpa) with respect to ply angle (y) and parameter (b) for the lipped-channel section. Table 2 Buckling stress scr (MPa) vs. parameter b.

0

10

20

30

40 50 θ [degree]

60

70

80

90

Fig. 19. Buckling stress scr (Mpa) with respect to ply angle (y) and parameter (g) for the lipped-channel section.

4. Load-deflection analysis

Hat-section 0 40.7 351

0.05 49.5 351

0.1 72.4 251

0.15 110.0 51

0.2 122.4 51

0.25 118.9 101

0.3 107.8 11

Lipped-channel-section 0 b scr (max) MPa 41.9 351 y (max)

0.05 52.9 351

0.1 75.9 301

0.15 104.4 151

0.2 123.1 51

0.25 136.2 121

0.3 125.9 201

b

scr (max) MPa y (max)

a ¼ 0.08, g ¼ 0.16.

Although eigenvalue analysis can capture the buckling load and mode shapes, it does not provide information about the postbuckling behaviour of the composite column. Hilburger and Starnes [16] also indicated that the laminate stacking sequence influenced both the buckling and post-buckling behaviour. Therefore, we conducted load-deflection analysis to investigate the post-buckling behaviour of the composite column. The effect of local buckling on the load-carrying capacity of the column was

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Table 3 Maximum buckling stress scr (MPa) vs. parameter g.

120

Hat-section

100

g scr (max)

0 84.7

0.08 0.16 0.24 0.32 0.4 0.5 0.6 0.8 1 115.6 118.9 119.8 121.1 121.6 123.5 126.7 136.8 147.0 101

101

101

101

101

101

101

101

σcr [Mpa]

22.81 101

Lipped-channel-section g 0 0.08 0.16 0.24 0.32 0.4 0.5 0.6 0.8 1 scr (max) 98.8 126.4 131.8 130.3 135.2 144.6 146.9 156.6 186.9 219.8 MPa 301 17.51 121 121 151 151 151 17.51 17.51 17.51 y (max)

Lipped-channel section

80

MPa

y (max)

1157

60

θ=0 θ = 10 θ = 20 θ = 30 θ = 40 θ = 45 θ = 50 θ = 60 θ = 70 θ = 80 θ = 90

bs = 0; α = 0.0 b1 = 12.5; β = 0.25 hs = 0; γ = 0.0

40 20

a ¼ 0.08, b ¼ 0.25.

0 0.0

120 θ=0 θ = 10 θ = 20 θ = 30 θ = 40 θ = 45 θ = 50 θ = 60 θ = 70 θ = 80 θ = 90

100 Hat section

60

20 0 0.0

50.0

100.0 λ

150.0

200.0

150.0

200.0

Fig. 22. Buckling stress scr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the hat section.

350 300

bs = 0; α = 0.0 b1 = 12.5; β = 0.25 hs = 0; γ = 0.0

40

100.0 λ

Lipped-channel section

250 σcr [Mpa]

σcr [Mpa]

80

50.0

200 150 100

Fig. 20. Buckling stress Icr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the hat section.

θ=0 θ = 10 θ = 20 θ = 30 θ = 40 θ = 45 θ = 50 θ = 60 θ = 70 θ = 80 θ = 90

bs = 12; α = 0.24 b1 = 12.5; β = 0.25 γ = 0.16 hs = 4;

50 0 0.0

350

Hat section

σcr [Mpa]

250 200 150

bs = 12; α = 0.24 b1 = 12.5; β = 0.25 hs = 4; γ = 0.16

100 50 0 0.0

50.0

100.0

150.0

200.0

λ Fig. 21. Buckling stress scr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the hat section.

100.0

150.0

200.0

λ Fig. 23. Buckling stress scr (Mpa) with respect to ply angle (y) and stiffener parameter (a) for the hat section.

140

Un-stiffened section Θ = 0° Θ = 10° Θ = 20° Θ = 22.8° Θ = 30° Θ = 45° Θ = 60°

120 Nominal stress [Mpa]

θ=0 θ = 10 θ = 20 θ = 30 θ = 40 θ = 45 θ = 50 θ = 60 θ = 70 θ = 80 θ = 90

300

50.0

100 80 60

Stiffened section Θ = 0° Θ = 10° Θ = 20° Θ = 22.8° Θ = 30° Θ = 45° Θ = 60°

40 20 0

also studied. The analysis was carried out using the RIKS method available in ABAQUS [22]. The initial imperfection was not applied to the analysis model. In this analysis the sections in Figs. 3 and 4 with a ¼ 0.08, b ¼ 0.25 and g ¼ 0.16 were analyzed. The column length was L ¼ 650 mm. The load-deflection curves for the hat-section column with various ply angles (y) are presented in Fig. 24. Since the curves corresponding to y4601 have a very small post-buckling load,

0

0.5

1 1.5 2 2.5 Axial displacement of loading point [mm]

3

Fig. 24. Load-deflection curves of the hat-section columns under axial compression.

they are not presented herein. The stiffness of the columns significantly decreased when the ply angle (y) increased. The postbuckling equilibrium paths were unstable for a ply angle (y) larger

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160 Un-stiffened Stiffened section section Θ = 0° Θ = 0° Θ = 10° Θ = 10° Θ = 12° Θ = 12° Θ = 20° Θ = 20° Θ = 30° Θ = 30° Θ = 45° Θ = 45° Θ = 60° Θ = 60°

Nominal stress [Mpa]

140 120 100 80 60 40 20 0 0

0.5

1 1.5 2 2.5 3 Axial displacement of loading point [mm]

3.5

Fig. 25. Load-deflection curves of the lipped-channel-section columns under axial compression.

than the transition angle (y4101 and yZ22.81 for stiffened and un-stiffened sections, respectively). Flexural–torsional buckling mode often has a less stable post-buckling path. However, the analysis shows that the composite column in this study had unstable post-buckling paths. This exhibits a characteristic of the composite material with respect to the buckling phenomena. The column with the hat section made of angle-ply material was sensitive to flexural–torsional buckling and its stiffness was reduced at quicker rate following buckling. The ply angle y ¼ 01 gave the highest post-buckling load for both the stiffened and unstiffened sections. The highest post-buckling stress of the stiffened section was 13% larger than the un-stiffened section. For the un-stiffened section, the highest post-buckling stress was 42% larger than the maximum buckling stress obtained in eigenvalue analysis. The post-buckling deformation shape of both the stiffened and un-stiffened columns was flexural–torsional mode at y4301, combined mode at yr301. The combined mode was a combination between local, flexural–torsional and distortional modes. The typical combined mode shapes are shown in Fig. 26.

Fig. 26. Combined deformation shapes of the composite columns of the hat section under axial compression obtained by geometric nonlinear analysis. (a) Local flexural–torsional shape of the column of the un-stiffened section, y ¼ 101. (b) Local flexural–torsional shape of the column of the stiffened section, y ¼ 101. (c) Flexural–torsional distortional shape of the column of the stiffened section, y ¼ 301.

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1159

Fig. 27. Deformation shape of the composite columns of the lipped-channel section under axial compression obtained by geometric nonlinear analysis. (a) Local-flexural shape of the column of the un-stiffened section, y ¼ 121. (b) Local flexural distortional shape of the column of un-stiffened section, y ¼ 301. (c) Local–torsional distortional shape of the column of the stiffened section, y ¼ 451.

The load-deflection curves of the lipped-channel-section column with various ply angles are shown in Fig. 25. The postbuckling paths of the stiffened section were unstable for all ply angles. Meanwhile, the post-buckling paths of the un-stiffened section were unstable at yZ301. The ply angle y ¼ 121 gives the highest post-buckling stress for both the stiffened and un-stiffened sections. For the un-stiffened section, the highest post-buckling stress was 45.8% larger than the maximum buckling stress obtained in eigenvalue analysis. Particularly, the post-buckling stress of the un-stiffened section was higher than the stiffened section for all ply angles. The post-buckling deformation shape of both the stiffened and un-stiffened section columns was flexural–torsional mode at y4601, combined mode at yr601. The typical combined mode shapes are shown in Fig. 27.

5. Conclusions The buckling and post-buckling of the thin-walled composite columns of the hat sections and lipped-channel sections composed of symmetric angle-ply laminate were investigated in this study. The sections of the width b ¼ 50 mm and height h ¼ 25 mm were analyzed. Non-dimensional geometric parameters a, b, g and l were defined to characterize the width, the height of the web stiffener, and the width of the lip and the length of the column, respectively. Columns of L ¼ 650 mm were analyzed for a, b, g, and the columns for a ¼ 0, 0.24, b ¼ 0.25, g ¼ 0, 0.16 were analyzed at different column lengths. Eigenvalue analyses were performed to

investigate the effects of the geometric parameters on the buckling behaviour of the columns. Load-deflection analyses were carried out to capture the post-buckling behaviour of the columns. The results of this study allow for several conclusions: (1) Flexural–torsional buckling mode had a lower critical stress than flexural mode for hat-section columns and vice versa for lipped-channel-section columns. Flexural buckling mode changes to distortional buckling mode at y ¼ 451. (2) The web stiffener was able to increase the local buckling stress of the composite column up to 2.5 times. The local buckling stress and number of halfwaves for two types of section shapes were almost identical. (3) The transition ply angle where the buckling mode changes from local to global was determined as 101 and 22.81 for the hat section with and without web stiffener, respectively, and 121 and 301 for the lipped-channel section with and without web stiffener, respectively. The maximum buckling stress was obtained at the transition angles and the minimum was at y ¼ 901. (4) The width of the stiffener (bs) had significant effect on the local and flexural–torsional buckling modes, but not much on distortional and flexural buckling modes. It increased the buckling stress up to 64% and 39% for the hat section and lipped channel section, respectively. The lip of the section significantly affected both local and distortion modes. It increased the maximum buckling stress of the column up to 3 and 3.2 times for the hat section and lipped-channel section, respectively. The height of the stiffener rapidly increased the

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buckling stress when it was greater than a half of the section height (g40.5). It influenced the flexural buckling mode more than flexural–torsional mode. The columns might have high buckling stress when the slenderness ratio was smaller than 50 and the ply angle is smaller than 301. Other ply angles gave high buckling stress only for very short column (lo12). (5) The highest post-buckling stress of the column with the unstiffened section was 42% and 45.8% greater than the maximum buckling stress obtained in eigenvalue analysis for the hat section and lipped-channel section, respectively. (6) The post-buckling paths of the columns with the hat section were unstable when the ply angles are yZ22.81 for the unstiffened section, and y4101 for the stiffened section. The post-buckling paths of the columns with the lipped-channel section were unstable at all ply angles for stiffened sections but at yZ301 for un-stiffened. Acknowledgement This work was supported by the Korean Ministry of Education, Science and Technology under the Brain Korea 21 Project. References [1] Turvey GJ, Wittrick WH. The influence of orthotropy on stability of some multi-plate structures in compression. Aeronaut Q 1973;24(1):1–8. [2] Lee DJ. The local buckling coefficient for orthotropic structural sections. Aeronaut J 1978:575. [3] Upadhya AR, Loughlan J. The effect of mode interaction in orthotropic fiberreinforced composite plain channel section columns. Compos Struct 1981:366–82. [4] Bauld RB, Tzeng L. A Vlasov theory for fiber-reinforced beams with thinwalled open cross-sections. Int J Solids Struct 1984;20(3):277–94. [5] Reddy AD, Rehfield LW, Bruttomesso RI. Local buckling and crippling of thinwalled composite structures under axial compression. J Aircr 1989;26:97–102.

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