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ISSN : 2093-5145(Print)
ISSN : 2288-0232(Online)
Journal of the Korean Society for Advanced Composite Structures Vol.10 No.5 pp.16-23
DOI : https://doi.org/10.11004/kosacs.2019.10.5.016

Flexural and Buckling Analyses of Shear-flexible Laminated Composite T-beams

Jae Sung Kim1, Sung Yong Back2
1Ph. D. Candidate, Department of Civil Engineering, Inje University, Gimhae, Korea
2Professor, School of Civil and Urban Engineering, Inje University, Gimhae, Korea

본 논문에 대한 토의를 2019년 11월 30일까지 학회로 보내주시면 2019년 12월호에 토론결과를 게재하겠습니다.


Corresponding author: Back, Sung Yong School of Civil and Urban Engineering, Gimhae Inje University, 197 Inje-ro, Gimhae, Gyeongsangnam-do, Republic of Korea Tel. +82-55-320-3433 Fax. +82-55-321-3410, E-mail. civsyb@inje.ac.kr
August 27, 2019 September 2, 2019 September 5, 2019

Abstract


This paper presents a shear-flexible laminated composite beam element for the flexural and buckling analyses of laminated composite T-beams by adopting an orthogonal Cartesian coordinate system. The proposed beam elements include flexural shear and warping deformation and all coupling terms due to material anisotropy adopting the first-order shear deformation beam theory. A displacement-based one-dimensional finite element model, namely, twonode, three-node, and four-node beam elements is developed to solve the governing equations of thin-walled composite beams. The derived geometric block stiffness of axially loaded composite T-beams is used to compute the critical buckling load of composite beams. The performance of the proposed beam element is tested through the deformation and buckling analyses and the obtained results are compared with the shell finite element results using ABAQUS. The influence of boundary conditions, span-to-height ratio, and web fiber orientation on the maximum displacement and critical buckling load of symmetric angle-ply composite beams is investigated.



전단변형을 고려한 적층복합 T형 보의 휨 및 좌굴해석

김 재성1, 백 성용2
1인제대학교 토목공학과 박사과정
2인제대학교 토목도시공학부 교수

초록


본 연구에서는 T형 박벽 보의 휨과 좌굴해석을 위해 직교좌표계에 근거한 전단변형을 고려한 적층복합 보 요소를 제안한다. 1차 전단변형 보 이론을 사용하여 유도된 유한요소는 휨 전단변형 및 뒴 비틀림과 재료 비등방성 성질에 따른 연계 성을 고려한다. 전체 포텐셜 에너지 원리를 이용해 지배방정식을 유도하였다. 지배방정식의 해를 구하기 위해 변위법에 근거한 2절점, 3절점, 4절점 요소의 세 가지 보 요소를 제안하였다. 압축력을 받는 T형 보의 기하학적 강성은 좌굴하중을 산정하는 데 사용된다. 제안된 보 요소를 검증하기 위해 처짐과 좌굴 해석을 수행하였으며 ABAQUS 상세 유한요소 해석결과와 비교하였다. 최대 처짐에 대한 파이버 방향성과 높이 대 지간 비율과 대칭 적층복합 보의 임계좌굴하중 영향을 조사하였다.



    1. INTRODUCTION

    Composite materials are extensively used in aerospace and civil industry engineering due to their characteristics of high strength-to-weight ratio, corrosion resistance, high energy absorption, and low thermal expansion among other construction materials. Because of their geometry and material properties, composite structures are susceptible to undergo large deformations and vulnerable to structural stability below the material strength.

    Many researchers have studied the problem of flexural-torsional buckling of composite beams during recent years. Bauld and Tzeng (1984) extended Vlasov’s beam theory (1961) of isotropic material to thin-walled laminated composite beams of open cross section. To overcome the discrepancy between classical beam theory and experimental results, a number of one-dimensional finite element methods incorporating shear deformation effects have been developed to predict the response of laminated composite beams (Maddur and Chaturvedi, 2000;Back and Will, 2008;Kim, 2011). Nevertheless, many of these have been restricted to I- and C-sections.

    Structural members of the T-section are usually used as bracing members or chord members of the truss and occur in the coped area of the girder. This member has a high degree of single symmetry and a relatively low torsional rigidity compared to other open cross-sections. Moreover, it is also well known that flexural deformation is coupled with torsional deformation. Therefore, this member may be subjected to flexural-torsional buckling than doubly symmetric beams. Lee (2003) performed an experimental and analytical study on the flexural-torsional buckling behavior of pultruded T-members under axial load. Back (2014) investigated flexural behavior of a laminated composite T-beam using a contour coordinate system.

    A shear-flexible finite element is proposed for the flexural and buckling analyses of laminate composite T-beams. The displacement fields at the mid surface of an element are defined based on the orthogonal Cartesian coordinate system. The seven governing equations of composite beams are established. Three beam elements, namely, two-node, three-node, and four-node elements, with seven degree-of-freedom per node have been developed. The geometric stiffness of axially loaded composite beams is also developed to predict critical buckling loads. FORTRAN program is written for the implementation of the proposed laminated composite T-beam. Parametric studies are performed to investigate the influence of boundary conditions, web fiber orientation and length-to-height ratio on the maximum displacement and buckling load of composite T-beams.

    2. VARIATIONAL FORMULATION

    Fig. 1 shows two sets of coordinate systems used in the present study. One is the orthogonal Cartesian coordinate system (x, y, z), and the other is the contour coordinate (x, n, s). The contour coordinates of an arbitrary point A are (r, q), in which r and q are the n- and s- coordinates of A, respectively.

    Based on thin-walled beam assumptions with Timoshenko’s beam theory, the longitudinal and transverse displacements of a mid-surface can be expressed as (Back and Will, 2008)

    u ¯ ( x , y , z ) = u ( x ) y θ z ( x ) + z θ y ( x ) ω ϕ x ( x ) υ ¯ ( x , y , z ) = υ ( x ) ( z z p ) θ x ( x ) w ¯ ( x , y , z ) = w ( x ) + ( y y p ) θ x ( x )
    (1)

    where u ( x ) , υ ( x ) , w ( x ) , and θ x ( x ) , θ y ( x ) , θ z ( x ) are the rigid body translations and rotations along the x, y, and z axes, respectively; ϕ x ( x ) is the change in twist about pole, and ( y p , z p ) is the coordinates of shear center shown in Fig. 2. ω is the warping function.

    The variation of the strain energy may be written as

    δ U = l [ N x δ x o + M y δ κ y + M z δ κ z + M ω δ κ ω + V z δ γ x z o + V y δ γ x y o + T δ γ ω o + M t δ κ x s ] d x
    (2)

    where x o , κ y , κ z , κ w , and κ x s are axial strain, curvatures in the y and z direction, warping curvature about principle pole, and twisting curvature, respectively. N x , M y , M z , M w , V y , V z , T , and M t are axial force, moments about the y and z axes, bimoment, shear force in the y and z directions, two contributions to the total twisting moment.

    The potential energy of a composite member subjected to a compressive axial load σ x o is written as (Bleich, 1952)

    Ω = 1 2 V σ x o [ ( υ ¯ ) 2 + ( w ¯ ) 2 ] d υ
    (3)

    Substituting Eq. (1) into Eqs. (2) and (3) and combining them, the weak form can be expressed as

    δ Π = l { N x δ u + M y δ θ y + M z δ θ z + M ω δ ϕ x + V z δ ( w + θ y ) + V y δ ( υ θ z ) + T δ ( θ x ϕ x ) + M t δ ( 2 θ x ) P ( υ δ υ + [ w + ( y p + e ) θ x ] ) δ w + [ ( y p + e ) w + ( I p / A e β z ) θ x ] δ θ x ) } d x
    (4)

    where Ip is the polar moment of inertia about pole, e is an eccentricity, and the Wagner’s coefficient βz is

    β z = [ A y 3 d A + A z 2 y d A ] / I z 2 y p
    (5)

    The stress-strain relationship for a kth orthotropic lamina of the flange can be written as (Back and Will, 2008).

    { σ x f τ x z f } k = [ Q ¯ 11 * f Q ¯ 16 * f Q ¯ 16 * f Q ¯ 66 * f ] k { x f γ x z f }
    (6)

    where for the plane stress assumptions (σs = 0), the condensed transformed reduced stiffness coefficients are given by

    Q ¯ 11 * f = Q ¯ 11 Q ¯ 12 2 / Q ¯ 22
    (7a)

    Q ¯ 16 * f = Q ¯ 16 Q ¯ 12 Q ¯ 26 / Q ¯ 22
    (7b)

    Q ¯ 66 * f = Q ¯ 66 Q ¯ 26 2 / Q ¯ 22
    (7c)

    Similar expressions for web may be easily obtained.

    Combining Eq. (6) with the resultant forces and moment, the force-displacement relationship for a laminated composite beam can be expressed as

    { N x M y M z M ω M t V y V z T } = { E 11 E 12 E 13 E 14 E 15 E 16 E 17 E 18 E 22 E 23 E 24 E 25 E 26 E 27 E 28 E 33 E 34 E 35 E 36 E 37 E 38 E 44 E 45 E 46 E 47 E 48 E 55 E 56 E 57 E 58 E 66 E 67 E 68 E 77 E 78 E 88 } { u θ y θ z θ x 2 θ x w + θ y υ θ z θ x ϕ x }
    (8)

    where Eij is the stiffnesses of composite beams. By performing proper through-thickness integration, we explicitly express non-zero stiffness for T-section:

    E 11 = A 11 β b β E 12 = B 11 w b 2 E 13 = A 11 β b β y p + A 11 w b 2 2 / 2 + B 11 f b 1 E 14 = B 11 w b 2 2 / 2 E 15 = B 16 β b β E 16 = A 16 f b 1 E 17 = A 16 w b 2 E 22 = A 11 f b 1 3 / 12 + D 11 w b 2 E 23 = B 11 w b 2 ( b 2 / 2 y p ) E 24 = B 11 f b 1 3 / 12 + D 11 w b 2 2 / 2 E 25 = D 16 w b 2 E 27 = B 16 w b 2 E 33 = A 11 β b β y p 2 + A 11 w b 2 2 ( y p + b 2 / 3 ) + b 1 ( 2 B 11 f y p + D 11 f ) E 34 = B 11 w b 2 2 ( y p / 2 b 2 / 3 ) E 35 = B 16 β b β y p B 16 w b 2 2 / 2 D 16 f b 1 E 36 = ( A 16 f y p B 16 f ) b 1 E 37 = A 16 w b 2 ( y p b 2 / 2 ) E 44 = D 11 f b 1 3 / 12 + D 11 w b 2 3 / 3 E 45 = D 16 w b 2 2 / 2 E 47 = D 16 w b 2 2 / 2 E 55 = D 66 β b β E 56 = B 66 f b 1 E 57 = B 66 w b 2 E 66 = A 66 w b 2 + A 66 f b 1 E 68 = A 55 w b 2 2 / 2 E 77 = A 55 f b 1 + A 66 w b 2 E 88 = A 55 f b 1 3 / 12 + A 55 w b 2 3 / 3
    (9)

    where a repeated index represents the sum; the index β takes the value 1 and 2, which denote the flange and the web, respectively. The symbols b1 and b2 represent the width of the flange and the web, respectively.

    The extensional, bending-extensional coupling, and bending stiffnesses in Eq. (9) are given by

    ( A i j f , B i j f , D i j f ) = Q ¯ i j ( 1 , n , n 2 ) d y    i , j = 1 , 2 , 6 ( A i j w , B i j w , D i j w ) = Q ¯ i j ( 1 , n , n 2 ) d z    i , j = 1 , 2 , 6 A i j = Q i j ¯ dz i , j = 4 , 5
    (10)

    Integrating the linear part of Eq. (4) by parts and collecting each displacement component, we can write the governing equations of thin-walled composite beams as follows:

    E 11 u + E 17 υ + E 16 w + ( E 18 + 2 E 15 ) θ x + E 12 θ y + E 16 θ y + E 13 θ z E 17 θ z + E 14 ϕ x E 18 ϕ x = p x E 16 u + E 67 υ + E 66 w + ( E 68 - 2 E 56 ) θ x + E 26 θ y + E 66 θ y + E 36 θ z E 67 θ z + E 46 ϕ x E 68 ϕ x = p x E 17 u + E 77 υ + E 67 w + ( E 78 - 2 E 57 ) θ x + E 27 θ y + E 67 θ y + E 37 θ z E 77 θ z + E 47 ϕ x E 78 ϕ x = p y ( E 18 2 E 15 ) u + ( E 78 2 E 57 ) υ + ( E 68 2 E 56 ) w + ( E 88 - 4 E 58 + 4 E 55 ) θ x + ( E 28 2 E 25 ) θ y + ( E 68 2 E 56 ) θ y + ( E 38 2 E 35 ) θ z + ( 2 E 57 E 78 ) θ z + ( E 48 2 E 45 ) ϕ x + ( 2 E 58 E 88 ) ϕ x = m x E 12 u E 16 u + E 27 υ E 67 υ + E 26 w E 66 w + ( E 28 2 E 25 ) θ x + ( 2 E 56 E 68 ) θ x + E 22 θ y E 66 θ y + E 23 θ z ( E 27 E 36 ) θ z + E 67 θ z + E 24 ϕ x ( E 28 + E 46 ) θ x + E 68 ϕ x = m y E 13 u E 17 u + E 37 υ E 77 υ + E 36 w E 67 w + ( E 28 2 E 35 ) θ x + ( 2 E 57 E 78 ) θ x + E 23 θ y + ( E 36 E 27 ) θ y E 67 θ y + E 33 θ z z 2 E 37 θ z + E 77 θ z + E 34 ϕ x ( E 38 + E 47 ) ϕ x + E 78 ϕ x = m z E 14 u + E 18 u + E 47 υ + E 78 υ + E 46 w + E 68 w + ( E 48 2 E 45 ) θ x + ( E 88 2 E 58 ) θ x + E 24 θ y + ( E 46 + E 28 ) θ y + E 68 θ y + E 34 θ z + ( E 38 E 47 ) θ z E 78 θ z + E 44 ϕ x + E 48 ϕ x E 88 ϕ x = b
    (11)

    Eq. (11) is a general equilibrium equation for composite beams that takes into account warping shear deformation and material anisotropy due to various types of loadings, including axial forces, shear forces, bending moments, torques, and bimoment. You can see that the governing equations for the seven variables are fully coupled. For negligible shear deformation, the seven governing equations can be easily reduced to the four equations given by Lee and Lee (2004).

    3. FINITE ELEMENT FORMULATION

    The same shape function is adopted for all translational and rotational displacements to derive three different beam elements; linear, quadratic, and cubic elements.

    u = α = 1 n N α u α , υ = α = 1 n N α υ α , w = α = 1 n N α w α , θ i = α = 1 n N α θ i α ( i = x , y , z ) , ϕ x = α = 1 n N α ϕ x α
    (12)

    where n and Nα are the number of nodes and the shape function of node α, respectively.

    The element displacement vector de may be written as nodal displacement vector dα:

    d e = [ d 1 , d 2 , , d n ] T d α T = [ u , υ , w , θ x , θ y , θ z , ϕ x ]
    (13)

    Substituting Eq. (12) into Eq. (4), the equilibrium equations for composite beams can be expressed as

    [ k e + k g ] d e = f e
    (14)

    where fe is the element force vector and the element linear and geometric stiffness matrices ke and kg can be written in block matrix form as

    k e = [ k 11 e k 12 e k 1 n e k 1 2 e k 22 e k 2 n e k 1 n e k 2 n e k n n e ]
    (15a)

    k g = [ k 11 g k 12 g k 1 n g k 1 2 g k 22 g k 2 n g k 1 n g k 2 n g k n n g ]
    (15b)

    in which the expression for any block κ α β e ( α , β = 1 , 2 , , n ) is given in Back and Will(2008).

    The linear buckling problem can be written as

    [ K E + λ K ¯ G ] δ d = 0
    (16)

    where λ is the load factor for the reference load and K ¯ G is the initial geometric stiffness matrix at the reference load.

    4. NUMERICAL EXAMPLES

    Numerical results were made to validate the flexural analysis of laminated composite T-beams. In this study, two-node, three-node and four-node beam elements are considered. Then, apply the current model to the buckling analysis of composite beams and investigate the influence of aspect ratio and web fiber orientation on the critical buckling loads of composite T-beams.

    In the following numerical analysis, the cross section shown in Fig. 2 is used: b 1 = 31.47 mm , b 2 = 71.86 mm . The total thickness of flange and web is 3.139mm and 2.192mm, respectively. The material of the beam is made of glass epoxy with the following engineering constants: E 11 = 53.78 GPa , E 22 = E 33 = 17.93 GPa, G 12 = G 13 = 8.96 GPa , G 23 = 3.45 GPa , ν 12 = ν 13 = 0.25 , ν 23 = 0.34 .

    4.1 Static Analysis

    Composite beams with L/h = 10 and L/h = 20 under uniformly distributed load of 5 kN/m and 0.5 kN/m, respectively, throughout its length are performed for two types of boundary conditions: cantilever beam and simply supported beam. A total sixteen plies of the same thickness on the flange and the web are taken into account. The stacking sequence of the beam flange is unidirectional and the stem is assumed to be [ θ / θ ] 4 s . For two different L/h ratios, the maximum vertical displacements of simply supported and cantilever beams based on two different assumptions (σs = 0 and s = 0) are shown in Table 1 and Table 2, respectively. A total of 340 S9R5 shell elements are used for ABAQUS calculation, whereas two quadratic elements are used to model the beam. For both beams with L/h = 10 and L/h = 20, the present model based on plane stress assumption (σs = 0) is in a very good agreement with the ABAQUS results for the entire fiber angle range considered. On the other hand, the analysis based on s = 0 assumption seems to underestimate the maximum deflection by up to 15.5% at the sequence of lamination θ = 45o.

    In the next example, cantilever beams with L/h = 5 and L/h = 10 under a concentrated tip load of 1 kN are considered to investigate the effects of coupling and transverse shear deformation. Flange and stem are considered four layers of unidirectional and antisymmetric angle ply laminates [θ, - θ]2, respectively. The assumption of σs = 0 is made for every analysis. The variation in the maximum displacement of the beam with fiber angle is shown with the solutions based on classical beam theory (CBT) and first-order beam theory (FOBT) in Fig. 3.

    For convenience, the non-dimensionalized maximum displacement, ν ¯ = υ E 2 b 2 3 / ( P l 3 ) , is used. According to the first-order shear deformation theory, a closed-form solution of maximum transverse deflection of laminated beams can be computed for cantilever beams under the concentrated tip load P as follows (Reddy, 1997):

    ν max = P L 3 3 ( E I z ) c o m + P L ( G A z ) c o m
    (17)

    where the first and second terms denote the deflection due to pure bending (CBT) and the deflection due to shear deformation, respectively. The variation in bending and shear components of the vertical displacement with fiber orientation is shown in Fig. 4 for the ratio of L/h = 5. As the fiber angle changes, the bending component increases faster, while the shear one does not change significantly.

    It is seen from Fig. 3 that the influence of the shear deformation on the maximum deflection in this lay-up is small for L/h = 10. For lower span-to-height ratio (L/h = 5), however, the CBT solution excluding shear effect significantly underestimates the displacement over the entire range of fiber angles. Since the coupling stiffnesses E 13 , E 15 , E 27 , and E 35 do not vanish and all the other coupling stiffnesses are zero, the orthotropy solution given in Eq. (17) may not be accurate. However, due to the small coupling stiffness compared to the bending stiffness, the coupling effect due to material anisotropy becomes negligible for two different span-to-height ratios. As a result, this model closely matches with the orthotropic solution for this lay-up as shown in Fig. 3.

    4.2 Axially Loaded Cantilever Beams

    The cantilever beams with L/h = 10 and L/h = 20 are subjected to a compressive axial load at the pole. Both the flange and the web are made of sixteen plies and assume that they are symmetrically laminated about the mid-plane. Along with ABAQUS shell element results, critical buckling loads based on two different assumptions (σs = 0 and s = 0) are provided in Table 3 for various laminated stacking sequences. In the current finite element model, four quadratic elements are employed to represent the beam. It can be noticed that the present beam element based on plane stress assumption exhibits an excellent agreement with the ABAQUS solution for the entire fiber angle range under consideration. As expected, the longer span length reduces the critical buckling load and the lateral buckling becomes more pronounced. Columns with unidirectional fibers exhibit the highest critical buckling loads with respect to other laminations for all stacking sequences. As the ply angle θ increases, the critical buckling load is significantly reduced. While the current results based on σs = 0 are very consistent with the ABAQUS solution for all layups, an analysis based on plane strain assumption(s = 0) has been shown to overestimate the buckling load by up to 18.3% in the stacking sequence of θ = 45o. The critical buckling mode shape for the stacking sequence of [45/-45]4s is shown in Fig. 5. Disregarding all coupling stiffness, the orthotropic closed-form solution for cantilever boundary conditions is: P y = π 2 E 22 / 4 L 2 . The corresponding buckling load for the flexural mode in y direction yields Py = 976.7 N. This bucking load, ignoring the coupling stiffness, is slightly higher than the current result.

    The next example is identical to the previous example, except that the composite beam consists of four layers and the fiber angle is changed in two different ways: unidirectional fiber orientation of the flange and antisymmetric laminates [θ/ - θ]2 of the web (Case 1), and antisymmetric laminates [θ/ - θ]2 of the flange and unidirectional fiber orientation of the web (Case 2). For convenience, the following non- dimensional buckling load parameter is adopted: P ¯ c r = P c r L 2 10 2 / ( A E 2 b 2 2 ) . Buckling load parameters for cantilever beams with L/h = 10 and L/h = 20 are shown with ABAQUS results in Table 4. As seen from this table, the predictions in the present model are very consistent with the ABAQUS results. The buckling load in Case 1 is not significantly affected by the fiber angle of the web, while the buckling load decreases dramatically in Case 2 as the fiber angle in the flange changes. Columns with unidirectional fibers in the flange exhibit greater buckling loads than unidirectional fibers in the web. Also unlike the buckling load in the flexural-torsional behavior of an isotropic beam, the maximum buckling load of the anisotropic beam appears at θ = 0o.

    5. CONCLUSIONS

    Shear-flexible finite element models have been developed for flexural and buckling analyses of thin- walled laminated T-beams by adopting an orthogonal Cartesian coordinate system. The proposed beam elements include flexural shear and warping deformation and all coupling terms due to anisotropy. The seven governing equations are derived and for negligible shear deformations, the equilibrium equations are reduced to the four equations given in the literature. One-dimensional finite element model, two-node, three-node, four-node beam elements, have been developed. The numerical examples with the proposed element showed the ability to accurately predict the displacement and buckling load of composite T-beams. The effects of span-to-height ratio and fiber orientation on the maximum displacements and critical buckling load of symmetric angle ply composite beams were investigated. The results also showed that columns with unidirectional fibers in the flange exhibited greater buckling loads than unidirectional fibers in the web. This model has been found to be adequate and efficient for the flexural and buckling problems of laminated composite T-beams.

    Figure

    KOSACS-10-5-16_F1.gif
    Coordinates in Thin-walled Sections
    KOSACS-10-5-16_F2.gif
    Geometry of a T-beam
    KOSACS-10-5-16_F3.gif
    Variation of Vertical Displacement at Free End of CF Beam(L/h=5 and L/h=10)
    KOSACS-10-5-16_F4.gif
    Variation of the Bending and Shear Components of Vertical Displacement with Respect to Fiber Angle(L/h=5)
    KOSACS-10-5-16_F5.gif
    Buckling Mode Shape for a Cantilever Beam, [45/-45]4s

    Table

    Mid-span Displacement (cm) of a Simply Supported T-Beam
    Maximum Displacement(cm) of a Cantilever T-Beam
    Critical Buckling Loads of a Cantilever Beam(N)
    Comparison for Dimensionless Buckling Load Parameters of Cantilever Beams

    Reference

    1. ABAQUS/Standard User’s Manual (2003), Version 6.1, Hibbit, Kalsson & Sorensen Inc.
    2. Back, S. Y. and Will, K. M. (2008), “Shear-flexible Thin-walled Element for Composite I-beams,” Engineering Structures, Vol. 30, pp. 1447-1458.
    3. Back, S. Y. (2014), “Flexural Analysis of Laminated Composite T-beams,” Journal of Korean Society of Steel Construction, Vol. 26, No. 5, pp. 397-405 (in Korean).
    4. Bauld, N. R. and Tzeng, L. S. (1984), “A Vlasov Theory for Fiber-reinforced Beams with Thin-walled Open Cross Section,” International Journal of Solids and Structures, Vol. 20, No. 3, pp. 277-297.
    5. Bleich, F. (1952), Buckling Strength of Metal Structures, New York, McGraw-Hill.
    6. Kim, N. I. (2011), “Shear Deformable Doubly- and Monosymmetric Composite I-beams,” International Journal of Mechanical Sciences, Vol. 53, pp. 31-41.
    7. Lee, J. and Lee, S. (2004), “Flexural-torsional Behavior of Thin-walled Composite Beams,” Thin-Walled Structures, Vol. 42, No. 9, pp. 1293-1305.
    8. Lee, S. S. (2003), “Flexural-torsional Buckling of Pultruded T-sections,” Ph. D. Thesis, Georgia Institute of Technology, Atlanta, GA. USA.
    9. Maddur, S. S. and Chaturvedi, S. K. (2000), “Laminated Composite Open Profile Sections: Non-uniform Torsion of I-sections,” Composite Structures, Vol. 50, pp. 159-169.
    10. Reddy, J. N. (1997), Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press.
    11. Vlasov, V. Z. (1961), Thin-walled Elastic Beams, Jerusalem: Israel Program for Scientific Translations.