Flexural–torsional buckling of thin-walled I-section composites

Computers and Structures 79 (2001) 987±995

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Flexural±torsional buckling of thin-walled I-section composites Jaehong Lee a,*, Seung-Eock Kim b b

a Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea Department of Civil and Environmental Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea

Received 25 August 1999; accepted 31 August 2000

Abstract Buckling of an axially loaded thin-walled laminated composite is studied. A general analytical model applicable to the ¯exural, torsional and ¯exural±torsional buckling of a thin-walled I-section composite subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of ¯exural and torsional modes for arbitrary laminate stacking sequence con®guration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional ®nite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites addressing the e€ects of ®ber angle, anisotropy, and boundary conditions on the critical buckling loads and mode shapes of the composites. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thin-walled bar; I-section; Laminated composite; Classical lamination theory; Flexural±torsional buckling; Finite element method

1. Introduction Fiber-reinforced plastics (FRP) have been increasingly used over the past few decades in a variety of structures that require high ratio of sti€ness and strength to weight. In the construction industry, recent applications have shown the structural and cost eciency of FRP structural shapes, such as thin-walled open sections. The design of thin-walled members is often governed by stability considerations due to their slenderness. Thin-walled open section members made of isotropic materials have been studied by many researchers [1±7]. The di€erential equations for the coupled ¯exural±torsional buckling of open cross-section beams with iso-

*

Corresponding author. Tel.: +82-2-3408-3287; fax: +82-23408-3331. E-mail address: [email protected] (J. Lee).

tropic materials were derived by Chen and Atsuta [3], Chajes and Winters [4], and Yu [5]. Barsoum and Gallagher [6] presented a ®nite element model for coupled ¯exural torsion of thin-walled isotropic members. For isotropic materials, buckling by bending and torsion separately occur under axial load if the cross-section has two axes of symmetry [7]. For composite laminates, however, the bending and torsion are no longer uncoupled even for a doubly symmetric section, and ¯exural± torsional buckling should be considered. Little work have been done to address the buckling of composite thin-walled members. Bauld and Tzeng [8] extended Vlasov's thin-walled bar theory [1] to symmetric ®berreinforced laminates. Reh®eld and Atilgan [9] presented the buckling equations for uniaxially loaded composite open-section members. Barbero and Tomblin [10] conducted an experimental study on the Euler buckling of pultruded composite columns. Davalos and Qiao [11] presented a combined analytical and experimental evaluation of ¯exural±torsional and lateral±distorsional

0045-7949/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 0 ) 0 0 1 9 5 - 4

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J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

buckling of FRP composite wide-¯ange beams. Omidvar and Ghorbanpoor [12] developed a nonlinear ®nite element model for thin-walled open-section structural members made of laminated composites with symmetric stacking sequence. Kabir and Sherbourne studied the lateral buckling of thin-walled composite beams including shear e€ects [13], and local buckling e€ects [14]. In the present study, a general analytical model applicable to the ¯exural, torsional and ¯exural±torsional buckling of an I-section composite subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of ¯exural and torsional modes for arbitrary laminate stacking sequence con®guration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional ®nite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar with arbitrary boundary conditions. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites with angle ply and quasi-isotropic laminates. The e€ects of ®ber angle, material anisotropy, and boundary conditions on the critical buckling loads and mode shapes are parametrically studied.

Fig. 1. Geometry of thin-walled composite.

x a ˆ ya z

2. Kinematics The basic assumptions regarding the kinematics of thin-walled composites are stated as follows: · All laminates are linearly elastic and classical lamination theory is applicable. · Each plate member behaves as a thin-walled beam, and plane sections originally normal to the beam axis remains plane and normal to the beam axis after deformation. · Each laminate is thin and perfectly bonded. · Local buckling is not considered.

…2†

In Eq. (2), superscript a varies from 1 to 2 denoting top and bottom ¯anges. ya is location of the mid-surface of each ¯ange from the shear center as given by (Fig. 1) y1 ˆ y2 ˆ

b3 2

…3† b3 2

…4†

As shown in Fig. 1, a thin-walled composite beam with arbitrary laminate stacking sequence is considered. The resulting displacements U , V and W at a generic point in the cross-section of the thin-walled laminate are assumed to be of the form: U …x; y; z† ˆ u…x† yv0 …x† zw0 …x† V …x; y; z† ˆ v…x† z/…x† W …x; y; z† ˆ w…x† ‡ y/…x†

x/0 …x†

…1a† …1b† …1c†

where u, v and w are beam displacements at shear center in the x, y and z direction, respectively, and / is the angle of twist (Fig. 2). Function x is known as the warping function, and is given for I-shaped section as:

Fig. 2. Deformation con®guration in a thin-walled composite.

J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

The strains associated with the small-displacement theory of elasticity are given by x ˆ 0x ‡ zjy ‡ yjz ‡ xjx

…5†

where 0x , jy , jz and jx are axial strain, biaxial curvatures in the y and z direction, and warping curvature with respect to the shear center, respectively, de®ned by 0x ˆ u0 jy ˆ w00 jz ˆ v00

…6a† …6b† …6c†

/00

…6d†

jx ˆ

It is noted that the shear strains cxz and cyz become zero from the displacement ®elds given in Eqs. (1a)±(1c). However, the shear strains are generated from pure torsion action, and in order to account for this, additional displacement ®eld with respect to the shear center is introduced in the top and bottom ¯anges as a

a

a

a

U …x; y; z† ˆ u …x; z†

…y

ova …x; z† ya † ox

…y

ova …x; z† ya † oz

V …x; y; z† ˆ v …x; z† W a …x; y; z† ˆ wa …x; z†

…7a† …7b† …7c†

ya †jxs

…8†

where jxs is twisting curvature de®ned by jxs ˆ 2/0

…9†

For the web plate, the displacements are given as 3

3

U …x; y; z† ˆ u …x; y† V 3 …x; y; z† ˆ v3 …x; y†

ow3 …x; y† z ox 3 ow …x; y† z oy

W 3 …x; y; z† ˆ w3 …x; y†

ˆ

zjxs

where U  is the strain energy Z 1 U ˆ …rx x ‡ rxz cxz ‡ rxy cxy †dv 2 v

…13†

and V  is the potential of in-plane loads due to transverse de¯ection Z 1 V ˆ r0 ……V 0 †2 ‡ …W 0 †2 †dv …14† 2 v x where r0x is the constant in-plane edge axial stress. The variation of the strain energy is calculated by substituting Eqs. (5), (8) and (11) into Eq. (13) Z l  dU ˆ ‰Nx d0x ‡ My djy ‡ Mz djz ‡ Mx djx 0

‡ Mt djxs Šdx

…15†

where Nx , My , Mz and Mx are axial force, bending moments in the y and z direction, and warping moment (bimoment) with respect to the centroid, respectively, de®ned by integrating over the cross-sectional area A as: Z ‰Nx ; My ; Mz ; Mx Š ˆ rx ‰1; z; y; xŠdA …16† In Eq. (15), Mt is twisting moment by pure torsion de®ned by Z Mt ˆ ‰raxz …y ya † r3xy zŠdA …17† A

The variation of the potential of in-plane loads at centroid becomes by substituting displacement ®eld in Eq. (1a) into Eq. (14) " Z l …hk †2 …bk †2  0 ‡ rx bk hk v0 dv0 ‡ w0 dw0 ‡ dV ˆ 12 12 0 ! # ‡ …yk †2 /0 d/0 dx

…10a† …10b† …10c†

In Eqs. (10a)±(10c), superscript 3 denotes the web, u3 , v3 and w3 are the plate displacements at mid-surface of the web. In the same way as for the ¯ange, the shear strain in the web can be obtained as: c3xy

…12†

A

where superscript a denotes the top and bottom ¯anges as mentioned earlier. ua , va , and wa are the plate displacements at mid-surface of each ¯ange. From the displacement ®eld in the ¯ange as given in Eqs. (7a)±(7c) and the global displacement ®eld in Eq. (1b), the shear strain in the ¯ange is obtained as caxz ˆ …y

P ˆ U ‡ V 

989

…11†

3. Variational formulation The total potential energy of the system can be stated, in its buckled shape, as

…18†

where subscript k varies from 1 to 3, and repeated indices imply summation. bk and hk represent the width and the thickness of the ¯ange and web. The principle of total potential energy can be stated as dP ˆ d…U  ‡ V  † ˆ 0

…19†

Substituting Eqs. (15) and (18) into Eq. (19), and introducing the relationship r0x ˆ P 0 =A, the following weak statement is obtained: Z l 0ˆ Nx du0 My dw00 Mz dv00 Mx d/00 ‡ 2Mt d/0 0

  I0 ‡ P 0 v0 dv0 ‡ w0 dw0 ‡ /0 d/0 dx A

…20†

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J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

In Eq. (20), P 0 is the constant in-plane edge force de®ned by P0 ˆ

k

Z Nx ˆ

Z

where k is a buckling load parameter. In Eq. (20), I0 is polar moment of inertia of the cross-section about the centroid de®ned by

‡

I0 ˆ Iy ‡ Iz

‡

where Iy and Iz are second moment of inertia with respect to y and z axis, respectively de®ned by: Z Iy ˆ

A

Z Iz ˆ

A

…z†2 dA

…23a†

…y†2 dA

…23b†

The constitutive equations of a kth orthotropic lamina in the laminate coordinate system of ¯ange are given by rax raxz

"

k ˆ

a Q 11 a Q 16

a Q 16 a Q 66

#k 

ax caxz

 …24†

Similarly, the constitutive equations for web are given by 

r3x r3xy

"

k ˆ

3 Q 11 3 Q 16

3 Q 16 3 Q 66

#k 

3x c3xy

 …25†

It should be noted that ax ˆ 3x ˆ x in Eqs. (24) and  a and Q  3 represent transformed reduced (25), and Q ij ij sti€ness matrices of the ¯ange and web laminates. The transformed sti€nesses can be obtained from transformation of reduced sti€nesses to the ®ber angle direction [15]. The axial force Nx applied to the section of a thin-walled composite is a summation of the force in ¯ange, Nx1 and Nx2 and web, Nx3 , and can be expressed as Nx ˆ

Nx1

‡

Nx2

‡

Nx3

y1

b2 2 b2 2

Z

h

y1 ‡ 21

h3 2 h3 2

h1 2

 1 …0 ‡ zjy ‡ yjz ‡ xjx † fQ 11 x

 1 …y y1 †jxs gdy dz ‡Q 16 Z y2 ‡h22  2 …0 ‡ zjy ‡ yjz ‡ xjx † fQ y2

11

h2 2

x

 2 …y y2 †jxs gdy dz ‡Q 16 Z y3 ‡b23  3 …0 ‡ zjy ‡ yjz ‡ xjx † fQ y3

11

b3 2

x

 3 zjxs gdy dz Q 16

…27†

Similarly, the other stress resultants (My , Mz , Mx , Mt ) can also be written in terms of the generalized strains (0x , jy , jz , jx , jxs ). Consequently, the constitutive equations for a thin-walled laminated composite are obtained as 8 9 2 38 0 9 Nx > E11 E12 E13 0 E15 > x > > > > > > > > > > My = > 6 > > > E22 0 E24 E25 7 < 6 7< jy = 7 jz Mz ˆ 6 E 0 E 33 35 7> > …28† > > 6 > E44 0 5> > Mx > > 4 > jx > > > > > : ; : > ; Mt sym: E55 jxs where Eij are sti€nesses of the thin-walled composite, and can be de®ned by:

4. Constitutive equations



Z

b1 2

…21†

…22†

b1 2

…26†

By substituting Eqs. (16), (5), (8), (11), (24) and (25) into Eq. (26)

E11 ˆ bk Ak11

…29a†

E12 ˆ b3 B311 E13 ˆ ba Ba11 ‡ ya ba Aa11

…29b† …29c†

E15 ˆ ba Ba16

…29d†

b3 B316

b3a

Aa ‡ b3 D311 12 11 b3 E24 ˆ a ya Aa11 12 E25 ˆ b3 D316 E22 ˆ

…29e† …29f† …29g†

E33 ˆ ba Da11 ‡ 2ya ba Ba11 ‡ ya2 ba Aa11 ‡ E35 ˆ E44 ˆ

ba Da16 ‡ ya ba Ba16 b3a 2 a ya A11

b33

12

A311

…29h† …29i† …29j†

12 E55 ˆ bk Dk66

…29k†

where repeated indices imply summation. Akij , Bkij and Dkij matrices are extensional, coupling and bending sti€ness of the top and bottom ¯anges and web, respectively, de®ned by a na Z yk‡1 X …Aaij ; Baij ; Daij † ˆ …Qaij †k …1; y; …y†2 †dy …30† kˆ1

…A3ij ; B3ij ; D3ij † ˆ

yka

n3 Z X kˆ1

zk

zk‡1

…Q3ij †k …1; z; …z†2 †dz

…31†

where n1 , n2 and n3 denote the number of layers in the top and bottom ¯anges and web. yka is the local coordinate with respect to the mid-surface of each ¯ange.

J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

5. Governing equations for buckling The buckling equations of the present study can be derived by integrating the derivatives of the varied quantities by parts and collecting the coecients of du, dv, dw and d/: Nx0 ˆ 0 Mz00 My00

‡P v ˆ0 0

00

‡P w ˆ0

Mx00 ‡ 2Mt0 ‡ P 0

I0 00 / ˆ0 A

…36a†

…EIy †com ˆ E22

…36b†

…EIz †com ˆ E33

…36c†

…32b†

…EIx †com ˆ E44

…36d†

…32c†

…GJ †com ˆ 4E55

…36e†

…32d†

In Eqs. (35a)±(35d), it is well known that the three distinct buckling modes, ¯exural buckling in the y and z direction, and torsional buckling, are identi®ed in this case, and the corresponding buckling loads are given by the closed form for simply-supported boundary conditions [7]:

The natural boundary conditions are of the form: du : Nx ˆ Nx0

…33a†

dv : Mz0 ˆ Mz00

…33b†

dv0 : Mz ˆ Mz0

…33c†

dw : My0 ˆ My00

…33d†

dw0 : My ˆ My0 Mx0

d/ : 0

…33e† Mx00

‡ 2Mt ˆ

d/ : Mx ˆ

…33f†

Mx0

…33g†

where Nx0 , Mz00 , Mz0 , My00 , My0 , Mx00 and Mx0 are prescribed values. By substituting Eqs. (28) and (29a)±(29k) into Eqs. (32a)±(32d), the explicit form of the governing equations yield: E11 u00

E12 w000

000

iv

E13 v000 ‡ 2E15 /00 ˆ 0 000

E13 u

E33 v ‡ 2E35 / ‡ P v ˆ 0

E12 u000

E22 wiv

2E15 u000

2E35 v000

‡ 4E55 /00 ‡ P 0

…34a†

0 00

…34b†

E24 /iv ‡ 2E25 /000 ‡ P 0 w00 ˆ 0 2E25 w000

E24 wiv

…34c†

E44 /iv

I0 00 / ˆ0 A

…34d†

The above equations are most general form for ¯exural± torsional buckling of a thin-walled laminated composite with I-shaped section, and the dependent variables, u, v, w and / are fully coupled. If the stacking sequence of the web is symmetric, and the thin-walled composite is symmetric with respect to z axis, E12 ˆ E13 ˆ E15 ˆ E24 ˆ 0. Further, if both the web and ¯ange are balanced laminates, Aki6 ˆ Dki6 ˆ 0, and thus, E25 ˆ E35 ˆ 0. Finally, Eqs. (34a)±(34d) can be simpli®ed to the uncoupled di€erential equations as: …EA†com u00 ˆ 0

respect to y and z axis, …EIx †com and …GJ †com represent warping and torsional rigidities of the thin-walled composite, respectively, written as …EA†com ˆ E11

…32a† 0 00

991

p2 …EIy †com l2 2 p …EIz †com Pz ˆ 2 l 2  A p …EIx †com ‡ …GJ † P0 ˆ com I0 l2 Py ˆ

…35b†

…EIy †com wiv ‡ P 0 w00 ˆ 0   I0 00 …EIx †com /iv ‡ …GJ†com ‡ P 0 / ˆ0 A

…35c† …35d†

From above equations, …EA†com represents axial rigidity, …EIy †com and …EIz †com represent ¯exural rigidities with

…37b† …37c†

where Py , Pz and P0 are ¯exural buckling loads in the y and z direction, and torsional buckling load, respectively. 6. Finite element model The present theory for thin-walled composite beams described in the previous section was implemented via a displacement based ®nite element method. The generalized displacements are expressed over each element as a linear combination of the one-dimensional Lagrange interpolation function Wj and Hermite-cubic interpolation function wj associated with node j and the nodal values; uˆ

n X

uj Wj

…38a†

vj wj

…38b†

wj wj

…38c†

/j wj

…38d†

jˆ1

vˆ

n X jˆ1

wˆ

n X jˆ1

…35a†

…EIz †com viv ‡ P 0 v00 ˆ 0

…37a†

/ˆ

n X jˆ1

Substituting these expressions into the weak statement in Eq. (20), the ®nite element model of a typical element can be expressed as the standard eigenvalue problem: …‰KŠ

k‰GŠ†fDg ˆ f0g

…39†

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J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

where ‰KŠ is the element 2 K11 K12 K13 6 K22 0 6 ‰KŠ ˆ 4 K33 sym:

sti€ness matrix 3 K14 K24 7 7 K34 5 K44

and ‰GŠ is the element geometric sti€ness matrix 2 3 0 0 0 0 6 G22 0 0 7 7 ‰GŠ ˆ 6 4 G33 0 5 sym: G44 The explicit form of ‰KŠ and ‰GŠ are given by Z l E11 W0i W0j dx Kij11 ˆ 0 Z l Kij12 ˆ E13 W0i w00j dx 0 Z l Kij13 ˆ E12 W0i w00j dx 0 Z l Kij14 ˆ 2 E15 W0i w0j dx 0 Z l E33 w00i w00j dx Kij22 ˆ 0 Z l Kij24 ˆ 2 E35 w00i w0j dx 0 Z l Kij33 ˆ E22 w00i w00j dx 0 Z l Kij34 ˆ …E24 w00i w00j 2E25 w00i w0j † dx 0 Z l …E44 w00i w00j ‡ 4E55 w0i w0j † dx Kij44 ˆ 0

G22 ij

ˆ

G44 ij ˆ

G33 ij Z 0

l

Z ˆ

l 0

w0i w0j dx

I0 0 0 w w dx A i j

…40†

…41†

…42a† …42b† …42c† …42d† …42e† …42f† …42g† …42h† …42i†

…43a† …43b†

In Eq. (39), fDg is the eigenvector of nodal displacements corresponding to an eigenvalue fDg ˆ fu v w /gT

…44†

7. Numerical results and discussion A thin-walled composite bar with I-section and length l ˆ 8 m is considered in order to investigate the e€ects of ®ber orientation, anisotropy and boundary conditions on the critical buckling loads and mode shapes of the composite (Fig. 3). The results are reported for the composite with simply supported and fully

Fig. 3. Example of thin-walled I-section composite under axial load.

clamped boundary conditions under axial load. The geometry of the I-section is (10  20  1 cm3 ) and the following material properties of the lamina are used: E1 ˆ 133:4 GPa; G12 ˆ 3:67 GPa;

E2 ˆ 8:78 GPa m12 ˆ 0:26

…45†

where subscripts 1 and 2 indicate ®ber direction and perpendicular to ®ber direction, respectively. As a ®rst numerical example, four layers with equal thickness are considered as a symmetric angle-ply laminate ‰‡h= h= h= ‡ hŠ in the ¯anges, and the ®bers are placed at 0° to the x-axis in the web (unidirectional). The next case is that the top and bottom ¯anges are unidirectional, and the web is four-layered symmetric angle-ply laminate. The third case is that both the ¯anges and web are symmetric angle-ply stacking sequence. For all the cases considered, the ¯ange and the web laminates are balanced and symmetric. In addition, the stacking sequences of the top and bottom ¯anges are symmetric with respect to z-axis, accordingly, all the coupling sti€nesses E12 , E13 , E15 , E24 , E25 , and E35 become zero as seen in previous section. Therefore, ¯exural buckling and torsional buckling are uncoupled, and the solution can be given in the closed-form as in Eqs. (37a)± (37c). The buckling loads of the three distinct modes, ¯exural in the y- and z-directions and torsional modes, by the ®nite element analysis are compared to those of the closed-form solution for a simply supported thinwalled composite with ®ber angle change in the ¯anges. Excellent agreements are made between two results with reference to Fig. 4. This is because of the fact that all the coupling sti€nesses vanish in this case, and thus, the

J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

Fig. 4. Variation of the critical buckling loads with respect to ®ber angle change in the ¯anges for a simply supported composite.

orthotropic closed-form solutions given in Eqs. (37a)± (37c) are suciently accurate in predicting buckling loads. Figs. 4±6 show the variation of critical buckling loads of a simply supported thin-walled composite with ®ber angle change for the three buckling mode shapes. It is seen that the buckling load for ¯exural mode in ydirection (Py ) is well below the other two types of buckling loads, i.e. buckling loads for ¯exural mode in

Fig. 5. Variation of the critical buckling loads with respect to ®ber angle change in the web for a simply supported composite.

993

Fig. 6. Variation of the critical buckling loads with respect to ®ber angle change in the ¯anges and web for a simply supported composite.

z-direction (Pz ) and torsional mode (P0 ) through the entire range of ®ber angle variation for all the three cases. In general Pz decreases as the ®ber angle increases for all the cases. In torsional buckling, the laminate torsional rigidity, …GJ †com , has an important e€ect on the load-carrying capacity. That is, D66 terms in ¯anges and web become important, thus, placing the ®ber angle at 45° leads to considerable improvement in the torsional buckling load. The composite with ®ber angle variation in ¯anges shows more rapid variation in buckling loads as the ®ber angle changes than that of ®ber angle variation in the web. The composite with ®ber angle variation in both ¯anges and web is most sensitive to the ®ber orientation (Fig. 6). The next example is the same as before except that in this case, boundary conditions are clamped. Figs. 7±9 show the variation of critical buckling loads of a clamped thin-walled composite with ®ber angle for three distinct buckling mode shapes. It is again observed that the buckling load for ¯exural mode in y-direction (Py ) is well below the other two types of buckling loads, i.e. buckling loads for ¯exural mode in z-direction (Pz ) and torsional mode (P0 ) through the entire range of ®ber angle variation for all the three cases. In torsional buckling, the laminate with ®ber angle at 45° exhibits highest buckling load as already shown in the simply supported composite. As compared to the examples of simply supported boundary conditions, it can be seen that the buckling load for torsional mode is less sensitive to ®ber angle variation than that of simply supported cases.

994

J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

Fig. 7. Variation of the critical buckling loads with respect to ®ber angle change in the ¯anges for a clamped composite.

Fig. 8. Variation of the critical buckling loads with respect to ®ber angle change in the web for a clamped composite.

In order to investigate the e€ects of anisotropy, a four-layered unsymmetric laminate with a combination of 0°, 90°, 45° and 45° ®ber orientation in the ¯ange and unidirectional ®ber direction in the web is considered. The thin-walled composite, however, is symmetric with respect to the global z-axis, that is the stacking sequence of the top and the bottom ¯anges is symmetric with each other. In this case, the coupling sti€nesses E12 , E13 , E15 , E24 and E25 become zero, but E35 does not

Fig. 9. Variation of the critical buckling loads with respect to ®ber angle change in the ¯anges and web for a clamped composite.

vanish because the stacking sequence in each ¯ange is unsymmetric and unbalanced. Accordingly, the ¯exural buckling in the y direction is uncoupled, but the ¯exural buckling in the z direction and the torsional buckling are coupled. The buckling load for ¯exural mode in y direction yields Py ˆ 1:473  104 N, and does not change for any combination of laminate stacking sequence. In Table 1, the lowest three ¯exural±torsional buckling loads are presented for various laminate stacking sequence. The laminate sti€ness E33 is almost same for any combination of stacking sequence, that is, the ¯exural buckling in the z direction might not be sensitive to the laminate stacking sequence. It can be seen that the laminates with o€-axis lamina …45°† in the core have considerably low torsional rigidities (E55 ) resulting in low buckling loads compared to the other. All the laminates with E55 ˆ 3:797  102 N show similar values of buckling loads regardless their laminate stacking sequence. The values of the torsional rigidities of the laminates with o€-axis lamina near surface …‰45=0=90= 45Š; ‰45=90=0= 45Š† are largest, and accordingly their buckling loads are highest.

8. Concluding remarks A one-dimensional ®nite element model was developed to study the ¯exural±torsional buckling of a composite with I-section. The model is capable of predicting

J. Lee, S.-E. Kim / Computers and Structures 79 (2001) 987±995

995

Table 1 Buckling loads for various laminate stacking sequence Bottom ¯ange

Laminate sti€ness (Nm2 ) E33

E55

E35

P1

P2

P3

‰0=45= 45=90Š ‰90=45= 45=0Š ‰45=90= 45=0Š ‰45=0= 45=90Š ‰0=45=90= 45Š ‰90=45=0= 45Š ‰45= 45=90=0Š ‰45= 45=0=90Š ‰0=90=45= 45Š ‰90=0=45= 45Š ‰45=0=90= 45Š ‰45=90=0= 45Š

2:061E ‡ 06 1:967E ‡ 06 1:975E ‡ 06 2:037E ‡ 06 2:053E ‡ 06 1:990E ‡ 06 1:983E ‡ 06 2:013E ‡ 06 2:045E ‡ 06 2:013E ‡ 06 2:029E ‡ 06 1:998E ‡ 06

1:867E ‡ 02 1:867E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 3:797E ‡ 02 5:728E ‡ 02 5:728E ‡ 02

3:912E ‡ 03 3:912E ‡ 03 8:019E ‡ 03 8:019E ‡ 03 7:628E ‡ 03 7:628E ‡ 03 4:107E ‡ 03 4:107E ‡ 03 3:716E ‡ 03 3:716E ‡ 03 1:174E ‡ 04 1:174E ‡ 04

1:211E ‡ 05 1:208E ‡ 05 2:132E ‡ 05 2:139E ‡ 05 2:160E ‡ 05 2:153E ‡ 05 2:288E ‡ 05 2:289E ‡ 05 2:300E ‡ 05 2:299E ‡ 05 2:868E ‡ 05 2:830E ‡ 05

1:811E ‡ 05 1:807E ‡ 05 2:576E ‡ 05 2:607E ‡ 05 2:639E ‡ 05 2:609E ‡ 05 2:806E ‡ 05 2:819E ‡ 05 2:849E ‡ 05 2:838E ‡ 05 2:960E ‡ 05 2:952E ‡ 05

2:837E ‡ 05 2:834E ‡ 05 3:257E ‡ 05 3:327E ‡ 05 3:336E ‡ 05 3:264E ‡ 05 3:162E ‡ 05 3:197E ‡ 05 3:224E ‡ 05 3:185E ‡ 05 3:949E ‡ 05 3:933E ‡ 05

Buckling loads (N)

accurate buckling loads as well as the buckling modes for various con®guration including boundary conditions and ®ber angle in ¯ange and web. All of the possible buckling modes including ¯exural in the y and z direction and torsional mode are considered. The e€ects of anisotropy, boundary conditions and the ®ber angle of ¯ange and web on buckling load and mode shape of composite are studied. Based on the above analytical developments and numerical results, the following conclusions are made: · The composite with ®ber angle change in the ¯ange shows more rapid variation in buckling loads as the ®ber angle changes than that of ®ber angle changes in the web. · For torsional buckling, the composite with ®ber angle near 45° yields the highest load-carrying capacity. · For locally unsymmetric, but globally symmetric laminates, the ¯exural±torsional buckling occurs due to coupling sti€ness. The laminates with o€-axis lamina near the surface in the ¯anges have the higher load-carrying capacities due to larger torsional rigidities. As a natural extension of this study, lateral±distorsional buckling of thin-walled composites awaits future research.

Acknowledgements This work presented in this paper was supported by funds of National Research Laboratory program (2000N-NL-01-C-162) from Ministry of Science and Technology in Korea. Authors wish to appreciate the ®nancial support.

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