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Flexural-torsional buckling of thin-walled composite box beams
Thuc Phuong Vo∗ and Jaehong Lee†
Department of Architectural Engineering, Sejong University98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Korea
(Dated: November 30, 2011)
Buckling of an axially loaded thin-walled laminated composite is studied. A general analyti-cal model applicable to the flexural, torsional and flexural-torsional buckling of a thin-walledcomposite box beam subjected to axial load is developed. This model is based on the classicallamination theory, and accounts for the coupling of flexural and torsional modes for arbitrarylaminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and variousboundary conditions. A displacement-based one-dimensional finite element model is developed topredict critical loads and corresponding buckling modes for a thin-walled composite bar. Gov-erning buckling equations are derived from the principle of the stationary value of total potentialenergy. Numerical results are obtained for axially loaded thin-walled composites addressing theeffects of fiber angle, anisotropy, and boundary conditions on the critical buckling loads and modeshapes of the composites.
Fiber-reinforced composite materials have been used over the past few decades in a variety of structures. Compositeshave many desirable characteristics, such as high ratio of stiffness and strength to weight, corrosion resistance andmagnetic transparency. Thin-walled structural shapes made up of composite materials, which are usually producedby pultrusion, are being increasingly used in many engineering fields. In particular, the use of pultruded compositesin civil engineering structures await increased attention.The theory of thin-walled closed section members made of isotropic materials was first developed by Vlasov [1]
and Gjelsvik [2]. Many researchers have shown that thin-walled bars are susceptible to instability in a variety ofmodes, but a few publications have dealt with buckling behavior of such members. Closed-form solution for flexuraland torsional buckling of isotropic thin-walled bars are found in the literature [3-4]. For composite thin-walled bars,the flexural and torsional buckling are fully coupled in general even for a doubly symmetric cross-section due totheir material anisotropy. Bhaskar and Librescu [5] focused on the flexural buckling of single-cell extension-twistcoupled beams under axial compression. The effects of direct transverse shear and the parasitic bending-transverseshear coupling as well as those of different boundary conditions and ply-angles were discussed. Shield and Morey[6] developed a new theory for analysis buckling of composite beams of open and closed cross section. The theorytook into account deformation in the plane of the cross section due to anticlastic curvature. Suresh and Malhotra[7] studied buckling of laminated composite thin walled rectangular box beam configurations. The effect of numberof layers, lay-up sequence and fiber angle on buckling load was analys for symmetric and anti-symmetric lay-ups.Recently, Cortinez and Piovan [8] presented the stability analysis of composite thin-walled beams with open or closedcross-sections. This model is based on the use of the Hellinger-Reissner principle, that considers shear flexibility ina full form, general cross-section shapes and symmetric balanced or especially orthotropic laminates. More recently,Piovan and Cortinez [9] developed a new theoretical model for the generalized linear analysis of composite thin-walledbeams with open or closed cross-sections. This model allows studying many problems of static’s, free vibrations withor without arbitrary initial stresses and linear stability of composite thin-walled beams with general cross-sections.In the present study, the analytical model developed by Lee and Kim [10] and Vo and Lee [11] is extended to
the buckling behavior of a thin-walled composite box beam. This model applicable to the flexural, torsional andflexural-torsional buckling of a thin-walled composite box beam subjected to axial load is developed. This model is
∗Graduate student
†Associate Professor, corresponding author; Electronic address: [email protected]
2
FIG. 1 Definition of coordinates in thin-walled closed sections
based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrarylaminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions.A displacement-based one-dimensional finite element model is developed to predict critical loads and correspondingbuckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of thestationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled compositesaddressing the effects of fiber angle, anisotropy, and boundary conditions on the critical buckling loads and modeshapes of the composites.
II. KINEMATICS
The theoretical developments presented in this paper require two sets of coordinate systems which are mutuallyinterrelated. The first coordinate system is the orthogonal Cartesian coordinate system (x, y, z), for which the x andy axes lie in the plane of the cross section and the z axis parallel to the longitudinal axis of the beam. The secondcoordinate system is the local plate coordinate (n, s, z) as shown in Fig.1, wherein the n axis is normal to the middlesurface of a plate element, the s axis is tangent to the middle surface and is directed along the contour line of thecross section. The (n, s, z) and (x, y, z) coordinate systems are related through an angle of orientation θ as defined inFig.1. Point P is called the pole axis, through which the axis parallel to the z axis is called the pole axis.To derive the analytical model for a thin-walled composite beam, the following assumptions are made:
1. The contour of the thin wall does not deform in its own plane.
2. The linear shear strain γsz of the middle surface is to have the same distribution in the contour direction as itdoes in the St. Venant torsion in each element.
3. The Kirchhoff-Love assumption in classical plate theory remains valid for laminated composite thin-walledbeams.
4. Each laminate is thin and perfectly bonded.
5. Local buckling is not considered.
According to assumption 1, the midsurface displacement components u, v at a point A in the contour coordinatesystem can be expressed in terms of a displacements U, V of the pole P in the x, y directions, respectively, and therotation angle Φ about the pole axis,
u(s, z) = U(z) sin θ(s)− V (z) cos θ(s)− Φ(z)q(s) (1a)
v(s, z) = U(z) cos θ(s) + V (z) sin θ(s) + Φ(z)r(s) (1b)
These equations apply to the whole contour. The out-of-plane shell displacement w can now be found from theassumption 2. For each element of middle surface, the shear strain become
γsz =∂v
∂z+∂w
∂s= Φ′(z)
F (s)
t(s)(2)
3
where t(s) is thickness of contour box section, F (s) is the St. Venant circuit shear flow.After substituting for v from Eq.(1) and considering the following geometric relations,
dx = ds cos θ (3a)
dy = ds sin θ (3b)
Eq.(2) can be integrated with respect to s from the origin to an arbitrary point on the contour,
w(s, z) = W (z)− U ′(z)x(s)− V ′(z)y(s)− Φ′(z)ω(s) (4)
where differentiation with respect to the axial coordinate z is denoted by primes (′); W represents the average axialdisplacement of the beam in the z direction; x and y are the coordinates of the contour in the (x, y, z) coordinatesystem; and ω is the so-called sectorial coordinate or warping function given by
ω(s) =
∫ s
s◦
[r(s)− F (s)
t(s)
]ds (5a)∮
i
F (s)
t(s)ds = 2Ai i = 1, ..., n (5b)
where r(s) is height of a triangle with the base ds; Ai is the area circumscribed by the contour of the i circuit. Theexplicit forms of ω(s) and F (s) for box section are given in Ref.[11].The displacement components u, v, w representing the deformation of any generic point on the profile section are
given with respect to the midsurface displacements u, v, w by the assumption 3.
u(s, z, n) = u(s, z) (6a)
v(s, z, n) = v(s, z)− n∂u(s, z)
∂s(6b)
w(s, z, n) = w(s, z)− n∂u(s, z)
∂z(6c)
The strains associated with the small-displacement theory of elasticity are given by
ϵs = ϵs + nκs (7a)
ϵz = ϵz + nκz (7b)
γsz = γsz + nκsz (7c)
where
ϵs =∂v
∂s; ϵz =
∂w
∂z(8a)
κs = −∂2u
∂z2; κz = −∂
2u
∂z2; κsz = −2
∂2u
∂s∂z(8b)
All the other strains are identically zero. In Eq.(8), ϵs and κs are assumed to be zero. ϵz, κz and κsz are midsurfaceaxial strain and biaxial curvature of the shell, respectively. The above shell strains can be converted to beam straincomponents by substituting Eqs.(1), (4) and (6) into Eq.(8) as
ϵz = ϵ◦z + xκy + yκx + ωκω (9a)
κz = κy sin θ − κx cos θ − κωq (9b)
κsz = 2χsz = κsz (9c)
where ϵ◦z, κx, κy, κω and κsz are axial strain, biaxial curvatures in the x and y direction, warping curvature withrespect to the shear center, and twisting curvature in the beam, respectively defined as
ϵ◦z = W ′ (10a)
κx = −V ′′ (10b)
κy = −U ′′ (10c)
κω = −Φ′′ (10d)
κsz = 2Φ′ (10e)
4
The resulting strains can be obtained from Eqs.(7) and (9) as
ϵz = ϵ◦z + (x+ n sin θ)κy + (y − n cos θ)κx + (ω − nq)κω (11a)
γsz = (n+F
2t)κsz (11b)
III. VARIATIONAL FORMULATION
The total potential energy of the system can be stated, in its buckled shape, as
Π = U + V (12)
where U is the strain energy
U =1
2
∫v
(σzϵz + σszγsz)dv (13)
After substituting Eq.(11) into Eq.(13)
U =1
2
∫v
{σz
[ϵ◦z + (x+ n sin θ)κy + (y − n cos θ)κx + (ω − nq)κω
]+ σsz(n+
F
2t)κsz
}dv (14)
The variation of strain energy can be stated as
δU =
∫ l
0
(Nzδϵz +Myδκy +Mxδκx +Mωδκω +Mtδκsz)ds (15)
where Nz,Mx,My,Mω,Mt are axial force, bending moments in the x and y directions, warping moment (bimoment),and torsional moment with respect to the centroid, respectively, defined by integrating over the cross-sectional areaas
Nz =
∫A
σzdsdn (16a)
My =
∫A
σz(x+ n sin θ)dsdn (16b)
Mx =
∫A
σz(y − n cos θ)dsdn (16c)
Mω =
∫A
σz(ω − nq)dsdn (16d)
Mt =
∫A
σsz(n+F
2t)dsdn (16e)
The potential of in-plane loads V due to transverse deflection
V =1
2
∫v
σ0z
[(u′)2 + (v′)2
]dv (17)
where σ0z is the averaged constant in-plane edge axial stress, defined by σ0
z = P 0/A. The variation of the potentialof in-plane loads at the centroid is expressed by substituting the assumed displacement field into Eq.(17) as
δV =
∫v
P 0
A
[U ′δU ′ + V ′δV ′ + (q2 + r2 + 2rn+ n2)Φ′δΦ′ + (Φ′δU ′ + U ′δΦ′)
[n cos θ − (y − yp)
]+ (Φ′δV ′ + V ′δΦ′)
[n cos θ + (x− xp)
]]dv (18)
In Eq. (18), the following geometric relations are used (Fig.1)
x− xp = q cos θ + r sin θ (19a)
y − yp = q sin θ − r cos θ (19b)
5
FIG. 2 Geometry of thin-walled composite box section
The principle of total potential energy can be stated as
δΠ = δ(U + V) = 0 (20)
Substituting Eqs.(15) and (18) into Eq.(20), the following weak statement is obtained
δΠ =
∫ l
0
[P 0
[δU ′(U ′ +Φ′yp) + δV ′(V ′ − Φ′xp) + δΦ′(Φ′ Ip
A+ U ′yp − V ′xp)
]+ NzδW
′ +MyδU′′ +MxδV
′′ +MωδΦ′′ − 2MtδΦ
]dz (21)
IV. CONSTITUTIVE EQUATIONS
The constitutive equations of a kth orthotropic lamina in the laminate co-ordinate system of box section are givenby {
σzσsz
}k
=
[Q∗
11 Q∗16
Q∗16 Q∗
66
]k {ϵzγsz
}(22)
where Q∗ij are transformed reduced stiffnesses. The transformed reduced stiffnesses can be calculated from the
transformed stiffnesses based on the plane stress assumption and plane strain assumption. More detailed explanationcan be found in Ref.[12]The constitutive equations for bar forces and bar strains are obtained by using Eqs.(11), (16) and (22)
Nz
My
Mx
Mω
Mt
=
E11 E12 E13 E14 E15
E22 E23 E24 E25
E33 E34 E35
E44 E45
sym. E55
ϵ◦zκyκxκωκsz
(23)
where Eij are stiffnesses of the thin-walled composite, and can be defined by
E11 =
∫s
A11ds (24a)
E12 =
∫s
(A11x+B11 sin θ)ds (24b)
E13 =
∫s
(A11y −B11 cos θ)ds (24c)
6
E14 =
∫s
(A11ω −B11q)ds (24d)
E15 =
∫s
(A16F
2t+B16)ds (24e)
E22 =
∫s
(A11x2 + 2B11x sin θ +D11 sin
2 θ)ds (24f)
E23 =
∫s
[A11xy +B11(y sin θ − x cos θ)−D11 sin θ cos θ
]ds (24g)
E24 =
∫s
[A11xω +B11(ω sin θ − qx)−D11q sin θ
]ds (24h)
E25 =
∫s
[A16
F
2tx+B16(x+
F sin θ
2t) +D16 sin θ
]ds (24i)
E33 =
∫s
(A11y2 − 2B11y cos θ +D11 cos
2 θ)ds (24j)
E34 =
∫s
[A11yω −B11(ω cos θ + qy) +D11q cos θ
]ds (24k)
E35 =
∫s
[A16
F
2ty +B16(y −
F cos θ
2t)−D16 cos θ
]ds (24l)
E44 =
∫s
(A11ω2 − 2B11ωq +D11q
2)ds (24m)
E45 =
∫s
[A16
F
2tω +B16(ω − Fq
2t)−D16q
]ds (24n)
E55 =
∫s
(A66F 2
4t2+B66
F
t+D66)ds (24o)
where Aij , Bij and Dij matrices are extensional, coupling and bending stiffness, respectively, defined by
(Aij , Bij , Dij) =
∫n
Qij(1, n, n2)dn (25)
It appears that the laminate stiffnesses Eij depend on the cross section of the composites. The explicit forms ofthem can be calculated for composite box section and given in Ref.[11].
V. GOVERNING EQUATIONS FOR BUCKLING
The buckling equations of the present study can be derived by integrating the derivatives of the varied quantitiesby parts and collecting the coefficients of of δU, δV, δW and δΦ
N ′z = 0 (26a)
M ′′y + P 0(U ′′ +Φ′′yp) = 0 (26b)
M ′′x + P 0(V ′′ − Φ′′xp) = 0 (26c)
M ′′ω + 2M ′
t + P 0(Φ′′ IpA
+ U ′′yp − V ′′xp) = 0 (26d)
The natural boundary conditions are of the form
δW : Nz = N0z (27a)
δU : M ′y =M
′0y (27b)
δU ′ : M ′y =M0
y (27c)
δV : M ′x =M
′0x (27d)
δV ′ : M ′x =M0
x (27e)
δΦ : M ′ω + 2Mt =M
′0ω (27f)
δΦ′ : Mω =M0ω (27g)
7
where N0z ,M
′0y ,M
0y ,M
′0x ,M
0x ,M
′0ω ,M
0ω are prescribed values. By substituting Eqs.(10) and (23) into Eq.(26) the
explicit form of the governing equations can be expressed with respect to the laminate stiffnesses Eij as
E11W′′ − E12U
′′′ − E13V′′′ − E14Φ
′′′ + 2E15Φ′′ = 0 (28a)
E12W′′′ − E22U
iv − E23Viv − E24Φ
iv + 2E25Φ′′′ + P 0(U ′′ +Φ′′yp) = 0 (28b)
E13W′′′ − E23U
iv − E33Viv − E34Φ
iv + 2E35Φ′′′ + P 0(V ′′ − Φ′′xp) = 0 (28c)
E14W′′′ + 2E15W
′′ − E24Uiv − 2E25U
′′′ − E34Viv − 2E35V
′′′ − E44Φiv + 4E55Φ
′′
+P 0(Φ′′ IpA
+ U ′′yp − V ′′xp) = 0 (28d)
Eq.(28) is most general form for flexural-torsional buckling of a thin-walled laminated composite with a box section,and the dependent variables, U , V , W and Φ are fully coupled. If all the coupling effects are neglected and the crosssection is symmetrical with respect to both x- and the y-axes, Eq.(28) can be simplified to the uncoupled differentialequations as
(EA)comW′′ = 0 (29a)
−(EIy)comUiv + P 0U ′′ = 0 (29b)
−(EIx)comViv + P 0V ′′ = 0 (29c)
−(EIω)comΦiv +[(GJ)com + P 0 Ip
A
]Φ′′ = 0 (29d)
From above equations, (EA)com represents axial rigidity, (EIx)com and (EIy)com represent flexural rigidities withrespect to x and y axis, (EIω)com represents warping rigidity, and (GJ)com, represents torsional rigidity of thethin-walled composite, respectively, written as
(EA)com = E11 (30a)
(EIy)com = E22 (30b)
(EIx)com = E33 (30c)
(EIω)com = E44 (30d)
(GJ)com = 4E55 (30e)
It is well known that the three distinct buckling modes, flexural buckling in the x and y direction, and torsionalbuckling, are identified in this case, and the corresponding buckling loads are given by the orthotropy solution forgeneral boundary conditions [4]
Px =π2(EIx)com
(Kxl)2(31a)
Py =π2(EIy)com
(Kyl)2(31b)
P0 =A
Ip
[π2(EIω)com(K0l)2
+ (GJ)com
](31c)
where Px, Py, P0 are flexural buckling loads in the x and y direction, and torsional buckling load. For simply-supportedand cantilever beams, Kx = Ky = K0 = 1 and Kx = Ky = K0 = 2, respectively.
VI. FINITE ELEMENT FORMULATION
The present theory for thin-walled composite beams described in the previous section was implemented via adisplacement based finite element method. The generalized displacements are expressed over each element as a linearcombination of the one-dimensional Lagrange interpolation function Ψj and Hermite-cubic interpolation function ψj
8
associated with node j and the nodal values
W =n∑
j=1
wjΨj (32a)
U =
n∑j=1
ujψj (32b)
V =n∑
j=1
vjψj (32c)
Φ =
n∑j=1
ϕjψj (32d)
Substituting these expressions into the weak statement in Eq.(18), the finite element model of a typical elementcan be expressed as the standard eigenvalue problem
([K]− λ[G]){∆} = {0} (33)
where [K] is the element stiffness matrix
[K] =
K11 K12 K13 K14
K22 K23 K24
K33 K34
sym. K44
(34)
and [G] is the element geometric stiffness matrix
[G] =
G11 G12 G13 G14
G22 G23 G24
G33 G34
sym. G44
(35)
The explicit forms of [K] and [G] are given by
K11ij =
∫ l
0
E11Ψ′iΨ
′jdz (36a)
K12ij = −
∫ l
0
E12Ψ′iψ
′′j dz (36b)
K13ij = −
∫ l
0
E13Ψ′iψ
′′j dz (36c)
K14ij =
∫ l
0
(2E15Ψ′iψ
′j − E14Ψ
′iψ
′′j )dz (36d)
K22ij =
∫ l
0
E22ψ′′i ψ
′′j dz (36e)
K23ij =
∫ l
0
E23ψ′′i ψ
′′j dz (36f)
K24ij =
∫ l
0
(E24ψ′′i ψ
′′j − 2E25ψ
′′i ψ
′j)dz (36g)
K33ij =
∫ l
0
E33ψ′′i ψ
′′j dz (36h)
K34ij =
∫ l
0
(E34ψ′′i ψ
′′j − 2E35ψ
′′i ψ
′j)dz (36i)
9
TABLE I Buckling loads with different stacking sequences and boundary conditions
All other components are zero. In Eq.(33), {∆} is the eigenvector of nodal displacements corresponding to aneigenvalue
{∆} = {W U V Φ}T (37)
VII. NUMERICAL EXAMPLES
For verification purpose, a composite box beam with length l = 4m and the cross section as shown in Fig.3a withdifferent stacking sequences and boundary conditions under axial load is considered. Plane stress assumption (σs = 0)is made in the analysis. The following material properties are used (Ref.[8])
The results using the present analysis are compared with previously available results in Table I. It is seen that theresults by the present finite element analysis are in good agreement with the solution in Ref.[8].In order to investigate the effects of fiber orientation, and boundary conditions on the buckling loads and the mode
shapes, a thin-walled composite box beam with the same cross section and the length l = 8m under axial load isconsidered. The results are reported for composite beam with simply supported and cantilever boundary conditions.The following engineering constants are used
E1/E2 = 25, G12/E2 = 0.6, ν12 = 0.25 (39)
10
FIG. 3 Thin-walled composite box beam and two stacking sequences
0
15
30
45
60
75
90
105
0 15 30 45 60 75 90
P
θ
Py (angle-ply for flanges only)Px (angle-ply for flanges only)
Py (angle-ply for flanges and webs)
⋆ ⋆
⋆
⋆
⋆⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆Px (angle-ply for flanges and webs)
⋆⋆
⋆
⋆
⋆
⋆⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆
FIG. 4 Variation of the critical flexural buckling loads of the weak and strong axis with respect to fiber angle change in thewebs and flanges for a simply supported composite beam
For convenience, the following nondimensional buckling load is used
P =Pl2
b31tE2(40)
A simply supported composite beam with two different stacking sequences is considered. Firstly, symmetric angle-ply laminate [θ/− θ]s in the flanges, while the webs are assumed unidirectional, (Fig.3b). Secondly, both the flangesand webs are assumed to be symmetric angle-ply stacking sequence, (Fig.3c). For two cases considered, the flangeand the web laminates are balanced symmetric and thus, all the coupling stiffnesses become zero. Accordingly, theflexural buckling and the torsional buckling are uncoupled, and the solution can be given in the orthotropy solutionas in Eqs.(31a)-(31c). The buckling loads of the three distinct modes, the flexural mode in the x- and y-direction andthe torsional mode, by the finite element analysis are compared to those of the orthotropy solution with fiber anglechange in the flanges and webs. Excellent agreements are made between two results as given in Figs.4 and 5. This isbecause of the fact that all the coupling stiffnesses vanish in these cases, and thus, the orthotropy solution given inEqs.(31a)-(31c) is sufficiently accurate in predicting buckling loads. It is seen that the buckling load for the torsionalmode is well above the other two types of buckling loads, i.e. Px and Py.The next example is a cantilever composite beam with the left and right webs are considered as [−θ2] and [θ2]
respectively, while the flanges are unidirectional. All coupling stiffnesses are zero, but E25 does not vanish due to
11
0
1500
3000
4500
6000
7500
0 15 30 45 60 75 90
P
θ
P0 (angle-ply for flanges only)P0 (angle-ply for flanges and webs)
FIG. 5 Variation of the critical torsional buckling loads with respect to fiber angle change in the webs and flanges for a simplysupported composite beam
unsymmetric stacking sequence of the webs. Accordingly, the flexural buckling in the x-direction is uncoupled, whereasthe flexural buckling in the y-direction and the torsional buckling are coupled. The critical buckling loads by thefinite element analysis and the orthotropy solution, which neglects the coupling effects of E25 , from Eqs.(31a)-(31c)are shown in Fig.6. For unidirectional fiber direction, the critical buckling loads by the finite element analysis exactlycorrespond to the flexural buckling loads in y-direction. The mode shape corresponding to the critical buckling loadwith fiber angle θ = 15◦ is illustrated in Fig.7. It can be seen that the critical buckling mode shape exhibits doublecoupling (the flexural mode in y-direction and the torsional mode). Due to the large coupling stiffnesses E25 in therange of (θ = 5◦ − 45◦), this mode become predominantly flexural y-direction mode, with a little contribution fromtorsion. Therefore, the results by finite element analysis and orthotropy solution show discrepancy in this range. Asthe fiber angle increases, the coupling stiffnesses E25 become small. Therefore, the critical buckling mode is purelyflexural y-direction mode as shown in Fig.8 and thus, the results by othotropy solution and finite element analysis areidentical.In order to investigate the coupling effect further, the top flange and the left web are considered as [0◦/−θ2/90◦] and
[0◦/θ2/90◦], while the bottom flange and the right web are [45◦4]. For this stacking sequence, the coupling stiffnesses
E12, E13, E15, E25 and E35 become no more negligibly small. The mode shape corresponding to the critical bucklingload with with fiber angle θ = 15◦ is illustrated in Fig.10. Relative measures of the flexural displacements and thetorsional rotation show that the mode is triply coupled mode (the flexural mode in the x- and y-directions and thetorsional mode). This fact explains as the fiber angle changes, the orthotropy solution and finite element analysissolution show remarkably discrepancy indicating the coupling effects become significant. As fiber angle increases,since the coupling stiffnesses decrease, the discrepancy becomes small. That is, the orthotropy solution is no longervalid for unsymmetrically laminated beams, and triply coupled flexural-torsional buckling should be considered evenfor a double symmetric cross-section.
VIII. CONCLUDING REMARKS
An analytical model was developed to study the flexural-torsional buckling of a laminated composite box beamunder axial load. The model is capable of predicting accurate buckling load as well as buckling mode shapes forvarious configuration including boundary conditions, laminate orientation of the composite beams. To formulate theproblem, a one-dimensional displacement-based finite element method is employed. All of the possible buckling modesincluding the flexural mode in the x and y-direction and the torsional mode, and fully coupled flexural-torsional modeare included in the analysis. The model presented is found to be appropriate and efficient in analyzing bucklingproblem of a thin-walled laminated composite beam under axial load.
12
0
2
4
6
8
10
0 15 30 45 60 75 90
P
θ
Py (orthotropy)P1 (FEM)⋆
⋆
⋆
⋆⋆
⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆
FIG. 6 Variation of the critical buckling loads with respect to fiber angle change in the webs for a cantilever composite beam
0
0.05
0.1
0.15
0.2
0.25
0 0.25 0.5 0.75 1
z/L
uv
⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆ϕ
FIG. 7 Mode shapes of the flexural and torsional components for P1 = 4.042 of a cantilever composite beams with the fiberangle 15◦ in the top flange and the left web
Acknowledgments
The support of the research reported here by Korea Ministry of Construction and Transportation through Grant2006-C106A1030001-06A050300220 is gratefully acknowledged.
References
[1] Vlasov, V. Z., Thin-walled elastic beams, 2nd Edition, Israel Program for Scientific Translation, Jerusalem, Israel, 1961.
13
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.25 0.5 0.75 1
z/L
uv
⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆ϕ
FIG. 8 Mode shapes of the flexural and torsional components for P1 = 1.603 of a cantilever composite beams with the fiberangle 75◦ in the top flange and the left web
0
0.5
1
1.5
2
2.5
0 15 30 45 60 75 90
P
θ
Py (orthotropy)P1 (FEM)
⋆
⋆
⋆
⋆⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆
FIG. 9 Variation of the critical buckling load with respect to fiber angle change in the top flange and the left web for a cantilevercomposite beam
[2] Gjelsvik, A., The theory of thin-walled bars, John Wiley and Sons Inc., New York, 1981.[3] Timoshenko, S. P. and Gere, J.M., Theory of elastic stability, McGraw-Hill, New York, 1963.[4] Trahair, N.S., Flexural-torsional buckling of structures, CRC Press, London, 1993.[5] Bhaskar, K., and Librescu, L., ”Buckling under axial compression of thin-walled composite beams exhibiting extension-twist
coupling,” Composite Structures, Vol.31, 1995, pp.203-212.[6] Shield, C.K and Morey T.A, ”Kinematic theory of open and closed section thin-walled composite beams,” Journal of
Engineering Mechanics, Vol.123, No. 10, 1997, pp.1070-1081.[7] Suresh, R. and Malhotra, S.K., ”Some studies on buckling of laminated composite thin walled box beams,” Composite
Structures, Vol.40, No. 3-4, 1998, pp.267-275.
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-0.05
0
0.05
0.1
0.15
0.2
0.25
0 0.25 0.5 0.75 1
z/L
uv
⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
⋆ϕ
FIG. 10 Mode shapes of the flexural and torsional components for P1 = 1.557 of a cantilever composite beam with the fiberangle 15◦ in the top flange and the left web
[8] Cortinez, V.H. and Piovan, M.T., ”Stability of composite thin-walled beams with shear deformability,” Computers andStructures, Vol.84, 2006, pp. 978-990.
[9] Piovan, M.T. and Cortinez, V.H., ”Mechanics of shear deformable thin-walled beams made of composite materials,” Thin-Walled Structures, Vol.45, 2007, pp. 37-62.
[10] Lee, J. and Kim, S., ”Flexural-torsional buckling of thin-walled I-section composites,” Computers and Structures, Vol.79,2001, pp.987-995.
[11] Vo, T.P. and Lee, J., ”Flexural-torsional behavior of thin-walled closed-section composite box beams,” Engineering Struc-tures, In Press, Vol.29, 2007
[12] Jones, R. M., Mechanics of composite materials, Hemisphere Publishing Corp., New York, 1975.
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