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Flexural analysis of thin-walled composite beams using shear-deformable beam theory
Composite Structures 70 (2005) 212–222 www.elsevier.com/locate/compstruct
Flexural analysis of thin-walled composite beams using shear-deformable beam theory Jaehong Lee
*
Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Korea Available online 25 September 2004
Abstract This paper presents a flexural analysis of I-shaped laminated composite beams. A general analytical model applicable to thinwalled I-section composite beams subjected to vertical load is developed. This model is based on the shear-deformable beam theory, and accounts for the flexural response of the thin-walled composites for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric. Governing equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for thin-walled composites under vertical loading, addressing the effects of fiber angle and span-to-height ratio of the composite beam. 2004 Elsevier Ltd. All rights reserved. Keywords: Thin-walled structures; Laminated composites; Flexural response; 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 stiffness and strength to weight. In the construction industry, recent applications have shown the structural and cost efficiency of FRP structural shapes, such as thin-walled open sections through pultrusion manufacturing process. Thin-walled open section members made of isotropic materials have been studied by many researchers [1,2]. Bauld and Tzeng [3] extended VlasovÕs thin-walled bar theory [1] to symmetric fiber-reinforced laminates to develop the linear and nonlinear theories for the bending and twisting of thin-walled composite beams. Davalos et al. [4] studied the bending response of pultruded composite beams with various I and box sections experimentally and analytically. They found that the micromechanics formulae can be used to predict the ply stiffnesses of the laminate. *
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0263-8223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.08.023
Ascione et al. [5] presented the statical behavior of fiber-reinforced polymer thin-walled beams taking into account the effects of shear deformation. They compare their results with VlasovÕs classical solution. Barbero [6] investigated the behavior of thin-walled composite beams in his monograph. Recently, Qin and Librescu [7] proposed shear-deformable analytical model for anisotropic thin-walled beams including nonclassical effects. More recently, Lee and Lee [8] presented analytical model which accounts for flexural–torsional behavior of I-section composite beams. They developed one-dimensional finite element model to investigate the flexural–torsional behavior of thin-walled composite beams. In this paper, the analytical model developed in [8] is extended by incorporating transverse shear deformation. This model is based on the first-order sheardeformable beam theory, and accounts for all the structural coupling coming from the material anisotropy. Warping effects are also included. Governing equations are derived from the principle of the stationary value of total potential energy. Numerical results
J. Lee / Composite Structures 70 (2005) 212–222
are obtained for thin-walled composites under vertical loading, addressing the effects of fiber angle and spanto-height ratio of the composite beam.
213
the midsurface shear strains in the contour can be expressed with respect to the transverse shear and warping shear strains. cnz ðs; zÞ ¼ cxz ðzÞ sin hðsÞ cyz ðzÞ cos hðsÞ þ cx ðzÞqðsÞ ð2aÞ
2. Kinematics This paper requires three sets of coordinate systems: an orthogonal Cartesian coordinate system (x, y, z), an orthogonal coordinate system (n, s, z), and a contour coordinate s along the profile of the section with its origin at any point O on the profile section. Three sets of coordinate systems are mutually interrelated and shown in Fig. 1. The n axis as shown in Fig. 1 is normal to the middle surface of a plate element, the s axis is tangent to the middle surface and is directed along the contour line of the cross section. The basic assumptions regarding the kinematics of thin-walled composites are state as follows: 1. The contour of the thin wall does not deform in its own plane. 2. Transverse shear strains cxz , cyz and warping shear cx are incorporated. It is assumed that they are uniform over the cross-sections. According to assumption 1, the midsurface displacement components u, v at a point A in the contour coordinate system can be expressed in terms of a displacements U, V of the pole P in the x, y directions, respectively, and the rotation angle U about the pole axis,
csz ðs; zÞ ¼ cxz ðzÞ cos hðsÞ þ cyz ðzÞ sin hðsÞ þ cx ðzÞrðsÞ ð2bÞ Further, it is assumed that midsurface shear strain in s n direction is zero (csn ¼ 0). From the definition of the shear strain, csz can also be given for each element of middle surface as: csz ðs; zÞ ¼
o w ov þ os oz
ð3Þ
After substituting for v from Eq. (1) and considering the following geometric relations, dx ¼ ds cos h
ð4Þ
dy ¼ ds sin h
ð5Þ
Eq. (3) can be integrated with respect to s from the origin to an arbitrary point on the contour. By compar ing Eqs. (2) and (3), the midsurface axial displacement w can then be obtained as ðs; zÞ ¼ W ðzÞ þ Wy ðzÞxðsÞ þ Wx ðzÞyðsÞ þ Wx ðzÞxðsÞ w ð6Þ where Wx, Wy and Wx represent rotations of the cross section with respect to x, y and x, respectively, given by:
uðs; zÞ ¼ U ðzÞ sin hðsÞ V ðzÞ cos hðsÞ UðzÞqðsÞ
ð1aÞ
Wy ¼ cxz U 0
ð7Þ
vðs; zÞ ¼ U ðzÞ cos hðsÞ þ V ðzÞ sin hðsÞ þ UðzÞrðsÞ
ð1bÞ
Wx ¼ cyz V 0
ð8Þ
Wx ¼ cx U0
ð9Þ
These equations apply to the whole contour. The out can now be found from of-plane shell displacement w the assumption 2. For each element of middle surface,
y
s n
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 assuming the first order variation of inplane displacements v, w through the thickness of the contour as:
P r O
q Θ
z
When the transverse shear effect is ignored, Eq. (9) degenerate to Wy = U 0 , Wx = V 0 and Wx = U 0 , and as a result, the number of unknown variables reduces to four leading to the Euler–Bernoulli beam model. The prime ( 0 ) is used to indicate differentiation with respect to z; and x is the so-called sectorial coordinate or warping function given by Z xðsÞ ¼ rðsÞ ds ð10Þ
uðs; z; nÞ ¼ uðs; zÞ
ð11aÞ
ðs; zÞ vðs; z; nÞ ¼ vðs; zÞ þ nw s
ð11bÞ
x
Fig. 1. Definition of coordinates in thin-walled open section.
214
J. Lee / Composite Structures 70 (2005) 212–222
ðs; zÞ ðs; zÞ þ nw wðs; z; nÞ ¼ w ð11cÞ z and w denote the rotations of a transverse where, w s z normal about the z and s axis, respectively. These functions can be determined by considering that the midsurface shear strains cnz is given by definition: cnz ðs; zÞ ¼
o w o u þ on oz
ð12Þ
can By comparing Eqs. (2) and (12), the function w z be written as ¼ Wy sin h Wx cos h Wx q w ð13Þ z Similarly, using the assumption that the shear strain can be csn should vanish at midsurface, the function w s obtained u ¼ o w s os
respect to the shear center, and twisting curvature in the beam, respectively defined as z ¼ W 0
ð18aÞ
jx ¼ W0x
ð18bÞ
jy ¼ W0y
ð18cÞ
jx ¼ W0x
ð18dÞ
jsz ¼ U0 Wx
ð18eÞ
The resulting strains can now be obtained with respect to the beam strain components from Eqs. (15) and (17) as z ¼ z þ ðx þ n sin hÞjy þ ðy n cos hÞjx þ ðx nqÞjx
ð14Þ
ð19aÞ csz ¼ cxz cos h þ cyz sin h þ cx r þ njsz
ð19bÞ
3. Strains
cnz ¼ cxz sin h cyz cos h þ cx q
ð19cÞ
The strains associated with the small-displacement theory of elasticity are given by dividing the midsurface strains and curvatures of the contour:
4. Variational formulation
s ðs; z; nÞ ¼ s ðs; zÞ þ n js ðs; zÞ
ð15aÞ
jz ðs; zÞ z ðs; z; nÞ ¼ z ðs; zÞ þ n
ð15bÞ
csz ðs; z; nÞ ¼ csz ðs; zÞ þ n jsz ðs; zÞ
ð15cÞ
cnz ðs; z; nÞ ¼ cnz ðs; zÞ þ n jnz ðs; zÞ
ð15dÞ
Total potential energy of the system is calculated by sum of strain energy and potential energy, P¼UþV
where s ¼
ov ; os
s ¼ j
ow s os
ð16aÞ
z ¼
o w ; oz
z ¼ j
ow z oz
ð16bÞ
sz ¼ j
ow ow z þ s; os oz
nz ¼ 0 j
ð16cÞ
All the other strains are identically zero. In Eq. (16), s and j s are assumed to be zero, and z , j z and j sz are midsurface axial strain and biaxial curvatures of the shell, respectively. The above shell strains can be converted to beam strain components by substituting Eqs. (1) and (11) into Eq. (16) z ¼ z þ xjy þ yjx þ xjx
ð17aÞ
z ¼ jy sin h jx cos h jx q j
ð17bÞ
sz ¼ jsz j
ð17cÞ
where z , jx, jy, jx and jsz are axial strain, biaxial curvatures in the x and y direction, warping curvature with
where U is the strain energy Z 1 U¼ ðrz z þ rsz csz þ rnz cnz Þ dv; 2 v
ð20Þ
ð21Þ
The strain energy is calculated by substituting Eq. (17) into Eq. (21) Z n 1 U¼ rz ½z þ ðx þ n sin hÞjy þ ðy n cos hÞjx 2 v þðx nqÞjx þ rsz ½cxz cos h þ cyz sin h þ cx r þ njsz o ð22Þ þrnz ½cxz sin h cyz cos h þ cx q dv The variation of strain energy, Eq. (22), can be stated as Z ln N z dz þ M y djy þ M x djx þ M x djx þ V x dcxz 0 o þV y dcyz þ T dcx þ M t djsz dz ð23Þ
dU ¼
where Nz, Mx, My, Mx, Vx, Vy, T and Mt are axial force, bending moments in the x and y directions, warping moment (bimoment), shear force in the x and y direction, and torsional moments, respectively, defined by integrating over the cross-sectional area A as: Z Nz ¼ rz ds dn ð24aÞ A
J. Lee / Composite Structures 70 (2005) 212–222
Z
My ¼
rz ðx þ n sin HÞ ds dn
ð24bÞ
Mx ¼
rz ðy n cos HÞ ds dn
ð29aÞ
ð29bÞ
ð24dÞ
¼ Q 16 Q12 Q26 Q 16 22 Q
ð24eÞ
2 66 Q26 ¼ Q Q 66 22 Q
ð29cÞ
ð24cÞ
A
Z
Mx ¼
rz ðx nqÞ ds dn
A
Vx ¼
Z
ðrsz cos h þ rnz sin hÞ ds dn A
Vy ¼
Z
and for free strain assumption, ðrsz sin h rnz cos hÞ ds dn
ð24fÞ
A
T ¼
Z
ðrsz r rnz qÞ ds dn
ð24gÞ
A
Mt ¼
for free stress assumption, 2 11 Q12 ¼ Q Q 11 22 Q
A
Z
215
Z
ð24hÞ
rsz n ds dn A
The variation of the work done by external force can be stated as Z l dV ¼ qdV dz ð25Þ 0
where q is transverse load. Using the principle which the variation of the total potential energy is zero, the following weak statement is obtained: Z ln 0¼ N z dW 0 þ M y dW0y þ M x dW0x þ M x dW0x 0
þV x dðU 0 þ Wy Þ þ V y dðV 0 þ Wx Þ þ T dðU0 þ Wx Þ o þM t dðU0 Wx Þ þ qdV dz ð26Þ
5. Constitutive equations of plate elements
11 ¼ Q Q 11
ð30aÞ
¼ Q 16 Q 16
ð30bÞ
66 ¼ Q Q 66
ð30cÞ
The constitutive relation for out-of-plane stress and strain is given by 55 cnz rnz ¼ Q ð31Þ In Eq. (24), the stress resultants Nz, Mx, My, Mx, Mx, Mt, Vx, Vy and T can now be expressed with respect to the generalized strains (z ; jy ; jx ; jx ; jsz ; cxz ; cyz ; cx ) by combining Eqs. (24), (28) and (19). Consequently, the constitutive equations for a thin-walled laminated composite are obtained as 38 9 9 2 8 E11 E12 E13 E14 E15 E16 E17 E18 > z > Nz > > > > > > > 7> > > 6 >j > >M > > > > E E E E E E E 7 6 > > > > y > y 22 23 24 25 26 27 28 > > > 7 6 > > > > > > > > > 7 6 > > > jx > Mx > E33 E34 E35 E36 E37 E38 7> > > > > > > 6 > > > > > > 7> = = 6 < 7 6 j E44 E45 E46 E47 E48 x x 7 ¼6 7 6 > > 6 > E55 E56 E57 E58 7> > > > > 6 > jsz > > Mt > > > > 7> > > > > > > 6 cxz > Vx > E66 E67 E68 7 > > > > > > > > 7 6 > > > > > > 7> 6 > > > > > > > > c V E E 5 4 y > 77 78 > yz > > > > > > > ; ; : : cx T sym: E88 ð32Þ
The constitutive equation for in-plane stresses and strains in an arbitrary z–s coordinate system is then written 8 9k 2 11 Q < rz = 4 12 r ¼ Q : s; 16 rsz Q
12 Q 22 Q 26 Q
8 9 16 3k < z = Q 26 5 s Q 66 : csz ; Q
ð27Þ
ij are transformed reduced stiffnesses [9] and are where Q made up of material property with respect to the each layer. The above constitutive equation can be simplified by using free stress in contour direction (rs = 0) or free strain in contour direction (s = 0) assumption as
rz rsz
"
k ¼
Q 11 Q 16
Q 16 Q 66
#k
z csz
ð28Þ
In Eq. (32), Eij are stiffnesses of the thin-walled composite. It appears that the laminate stiffnesses Eij depend on the cross section of the composites given by Z E11 ¼ A11 ds Zs E12 ¼ ðA11 x þ B11 sin hÞ ds Zs E13 ¼ ðA11 y B11 cos hÞ ds Zs E14 ¼ ðA11 x B11 qÞ ds Zs B16 ds E15 ¼ Zs A16 cos h ds E16 ¼ s
216
E17 ¼ E18 ¼ E22 ¼ E23 ¼
J. Lee / Composite Structures 70 (2005) 212–222
Z
E68 ¼
A16 sin h ds Zs
E77 ¼
A16 r ds Zs Zs
E24 ¼
2
2
ðA11 x þ 2B11 x sin h þ D11 sin hÞ ds ðA11 xy þ B11 ðy sin h x cos hÞ D11 sin h cos hÞ ds
E26 ¼ E27 ¼ E28 ¼ E33 ¼
Z
Zs Zs Zs Zs
ðA11 xx þ B11 ðx sin h qxÞ D11 q sin hÞ ds ðB16 x þ D16 sin hÞ ds ðA16 x þ B16 sin hÞ cos h ds ðA16 x þ B16 sin hÞ sin h ds
E35 ¼ E36 ¼ E37 ¼
Z
Zs Zs Zs
E78 ¼ E88 ¼
E44 ¼ E45 ¼ E46 ¼ E47 ¼ E48 ¼
Z
Zs Zs Zs Zs Zs
E56 ¼ E57 ¼ E58 ¼ E66 ¼
ðA66 r sin h þ A55 q cos hÞ ds ðA66 r2 þ A55 q2 Þ ds
ðA16 y B16 cos hÞ sin h ds ðA16 y B16 cos hÞr ds
The above expressions are general for arbitrary cross sections. For I-section shapes, the explicit expressions become: ¼ Ak11 bk ¼ B311 b3 ¼ Aa11 ba y a Ba11 ba ¼ Bk16 bk ¼ Aa16 ba ¼ A316 b3 ¼ Aa16 y a ba
bað3Þ þ D311 b3 12 bð3Þ E24 ¼ ½Aa11 y a Ba11 a 12 E25 ¼ D316 b3 E27 ¼ B316 b3 ð3Þ
ðA11 x2 2B11 xq þ D11 q2 Þ ds
a a E33 ¼ ½Aa11 y ð2Þ a 2B11 y a þ D11 ba þ
ðB16 x D16 qÞ ds
E34 ¼
ðA16 x B16 qÞ cos h ds ðA16 x B16 qÞ sin h ds ðA16 x B16 qÞr ds
Zs B66 cos h ds Zs B66 sin h ds Zs B66 r ds ðA66 cos2 h þ A55 sin2 hÞ ds
E35 E36 E37 E38
b3 3 B 12 11 a ¼ B16 y a ba Da16 ba ¼ ðAa16 y a Ba16 Þba ¼ A316 y 3 b3 ¼ ðAa16 y a Ba16 Þy a ba
s
ðA66 A55 Þ sin h cos h ds
ð35Þ
ð3Þ
a a E44 ¼ ½Aa11 y ð2Þ a 2B11 y a þ D11
E47 E55 E56 E57 E58 E66 E68 E77
bað3Þ b3 3 þ D 12 12 11
¼ B316 b3 y 3 ¼ Dk66 bk ¼ Ba66 ba ¼ B366 b3 ¼ Ba66 y a ba ¼ Aa66 ba þ A355 b3 ¼ Aa66 y a ba þ A344 y 3 b3 ¼ Aa55 ba þ A366 b3 ð2Þ
s
E67 ¼
b3 3 A 12 11
ð3Þ
D66 ds
Z
ð33Þ
where, Aij, Bij and Dij matrices are extensional, coupling and bending stiffness, respectively, defined by Z ij ð1; n; n2 Þ dn ðAij ; Bij ; Dij Þ ¼ Q ð34Þ
E22 ¼ Aa11
Z
Zs
Zs
ðA16 y B16 cos hÞ cos h ds
ðA11 yx B11 ðx cos h þ qyÞ þ D11 q cos hÞ ds
s
E55 ¼
Z
ðB16 y D16 cos hÞ ds
ðA11 y 2 2B11 y cos h þ D11 cos2 hÞ ds
s
E38 ¼
ðA66 sin2 h þ A55 cos2 hÞ ds
E11 E12 E13 E15 E16 E17 E18
ðA16 x þ B16 sin hÞr ds
s
E34 ¼
ðA66 r cos h A55 q sin hÞ ds
s
s
E25 ¼
Zs s
s
Z
Z
E88 ¼
Aa66 y að2Þ ba
Aa bð3Þ b ð2Þ þ 55 a þ A355 b3 y 3 þ 3 12 12
!
J. Lee / Composite Structures 70 (2005) 212–222
dWx : M x
By substituting Eqs. (32) and (18) into Eq. (36), the explicit form of the governing equations yield: E11 W 00 þ E16 U 00 þ E17 V 00 þ ðE15 þ E18 ÞU00 þ E12 W00y þ E16 W0y
y1
þ E13 W00x þ E17 W0x þ E14 W00x þ ðE18 E15 ÞW0x ¼ 0 x h3
b3
þ E36 W00x þ E67 W0x þ E46 W00x þ ðE68 E56 ÞW0x ¼ 0
y2
þ E37 W00x þ E77 W0x þ E47 W00x þ ðE78 E57 ÞW0x ¼ q
ð38cÞ
ðE15 þ E18 ÞW 00 þ ðE56 þ E68 ÞU 00 þ ðE57 þ E78 ÞV 00
Fig. 2. Geometry of a thin-walled composite.
where the superscript in the parenthesis ( ) denotes the power, and repeated index denotes summation. Index k varies from 1 to 3 whereas a varies from 1 to 2 implying top and bottom flanges (1, 2) and web (3) as shown in Fig. 2. bk denotes width of flanges and web. All the other Eij are zero.
þ ðE55 þ 2E58 þ E88 ÞU00 þ ðE25 þ E28 ÞW00y þ ðE56 þ E68 ÞW0y þ ðE56 þ E68 ÞW00x þ ðE57 þ E78 ÞW0x þ ðE45 þ E48 ÞW00x þ ðE88 E55 ÞW0x ¼ 0
ð38dÞ
E12 W 00 E16 W 0 þ E26 U 00 E66 U 0 þ E27 V 00 E67 V 0 þ ðE25 þ E28 ÞU00 ðE56 þ E68 ÞU0 þ E22 W00y E66 Wy þ E23 W00x þ ðE27 E36 ÞW0x E67 Wx þ E24 W00x
6. Governing equations
þ ðE28 E25 E46 ÞW0x þ ðE56 E68 ÞWx ¼ 0
The governing equations of the present approach can be derived by integrating the derivatives of the varied quantities by parts and collecting the coefficients of dW, dU, dV, dU, dWy, dWx and dWx: ð36aÞ N 0z ¼ 0 0 x
ð36bÞ
0 y
ð36cÞ 0
ð38bÞ
E17 W 00 þ E67 U 00 þ E77 V 00 þ ðE57 þ E78 ÞU00 þ E27 W00y þ E67 W0y b2
V ¼q
ð38aÞ
E16 W 00 þ E66 U 00 þ E67 V 00 þ ðE56 þ E68 ÞU00 þ E26 W00y þ E66 W0y
h2
V ¼0
ð37gÞ
y b1 h1
z
217
ðM t þ T Þ ¼ t
ð36dÞ
M 0y V x ¼ 0
ð36eÞ
M 0x V y ¼ 0
ð36fÞ
M 0x þ M t T ¼ 0
ð36gÞ
The natural boundary conditions are of the form: dW : N z
ð37aÞ
dU : V x
ð37bÞ
dV : V y
ð37cÞ
dU : T þ M t
ð37dÞ
dWy : M y
ð37eÞ
dWx : M x
ð37fÞ
ð38eÞ
E13 W 00 E17 W 0 þ E36 U 00 E67 U 0 þ E37 V 00 E77 V 0 þ ðE35 þ E38 ÞU00 ðE57 þ E78 ÞU0 þ E23 W00y þ ðE36 E27 ÞW0y E67 Wy þ E33 W00x E77 Wx þ E34 W00x þ ðE38 E35 E47 ÞW0x þ ðE57 E78 ÞWx ¼ 0
ð38fÞ
E14 W 00 þ ðE15 E18 ÞW 0 þ E46 U 00 þ ðE56 E68 ÞU 0 þ E47 V 00 þ ðE57 E78 ÞV 0 þ ðE45 þ E48 ÞU00 þ ðE55 E88 ÞU0 þ E24 W00y þ ðE46 þ E25 E28 ÞW0y þ ðE56 E68 ÞWy þ E34 W00x þ ðE47 þ E35 E38 ÞW0x þ ðE57 E78 ÞWx þ E44 W00x ðE88 2E58 þ E55 ÞWx ¼ 0
ð38gÞ
Eq. (38) are most general form for a thin-walled laminated composite beam. For general anisotropic materials, the dependent variables, U, V, W, U, Wx, Wy and Wx are fully-coupled implying that the beam undergoes a coupled behavior involving bending, twising, extension, transverse shearing, and warping. If all the coupling effects are neglected, Eq. (38) can be simplified to the uncoupled differential equations as: ðEAÞcom W 00 ¼ 0
ð39aÞ
ðGAy Þcom ðU 00 þ W0y Þ ¼ 0
ð39bÞ
ðGAx Þcom ðV 00 þ W0x Þ ¼ q
ð39cÞ
218
J. Lee / Composite Structures 70 (2005) 212–222
ðGJ 1 Þcom U00 ðGJ 2 Þcom W0x ¼ 0
ð39dÞ
ðEI y Þcom W00y ðGAy Þcom ðU 0 þ Wy Þ ¼ 0
ð39eÞ
ðEI x Þcom W00x ðGAx Þcom ðV 0 þ Wx Þ ¼ 0
ð39fÞ
ðEI x Þcom W00x þ ðGJ 2 Þcom U0 ðGJ 1 Þcom Wx ¼ 0
ð39gÞ
From above equations, (EA)com represents axial rigidity, (GAx)com, (GAy)com represent shear rigidities with respect to x and y axis, (EIx)com and (EIy)com represent flexural rigidities with respect to x and y axis, (EIx)com represents warping rigidity, and (GJ1)com, (GJ2)com represent torsional rigidities of the thin-walled composite, respectively, written as ðEAÞcom ¼ E11
ð40aÞ
ðEI y Þcom ¼ E22
ð40bÞ
ðEI x Þcom ¼ E33
ð40cÞ
ðEI x Þcom ¼ E44
ð40dÞ
ðGAy Þcom ¼ E66
ð40eÞ
ðGAx Þcom ¼ E77
ð40fÞ
ðGJ 1 Þcom ¼ E55 þ E88
ð40gÞ
ðGJ 2 Þcom ¼ E55 E88
ð40hÞ
For bending analysis with respect to x-axis, only Eqs. (39c) and (39f) are involved, and these equations are well-known Timoshenko beam equations.
7. Finite element model The present theory for thin-walled composite beams described in the previous section was implemented via a displacement based one-dimensional finite element method. The generalized displacements are expressed over each element as a linear combination of the onedimensional Lagrange interpolation function wj associated with node j and the nodal values: n X
W ¼
wj wj ;
U¼
j¼1
U¼
n X
uj wj ;
V ¼
j¼1
/j wj ;
j¼1
Wx ¼
n X
n X j¼1
Wy ¼
n X
Wx ¼
vj w j ;
j¼1
wyj wj ;
j¼1
wxj wj ;
n X
n X j¼1
wxj wj
ð41Þ
Substituting these expressions into the weak statement in Eq. (26), the finite element model of a typical element can be expressed as 38 9 8 9 2 K 11 K 12 K 13 K 14 K 15 K 16 K 17 > > > > >0> >w> > 7> 6 > > > > > > > > > > 7 6 > > > > 0 u K K K K K K > > > 22 23 24 25 26 27 7> 6 > > > > > > > > 7> 6 > > > > > > > > > > > 7 6 f v K K K K K > > > > 33 34 35 36 37 3 > > > 7> 6 7< = < = 6 7 6 ¼ / K K K K 0 44 45 46 47 7 6 > > > > > > > 7> 6 > > > > > > > 7> 6 w 0 K K K > > > > 55 56 57 y > > > 7> 6 > > > > > > > 7> 6 > > > > > > > > 7 6 > > > > 0 K K w 66 67 5> x > > 4 > > > > > > > > ; : ; : > 0 sym: K 77 wx ð42Þ where [K] is the element stiffness matrix given by
J. Lee / Composite Structures 70 (2005) 212–222
Z
l
E11 w0i w0j dz
K 37 ij
E17 w0i w0j dz
K 44 ij
E18 w0i w0j dz
K 45 ij
ðE15 þ E18 Þw0i w0j dz
K 46 ij
K 15 ij
Z lh i ¼ E12 w0i w0j þ E16 w0i wj dz
K 47 ij
K 16 ij
Z lh i ¼ E13 w0i w0j þ E17 w0i wj dz
K 17 ij
Z lh i ¼ E14 w0i w0j þ ðE18 E15 Þw0i wj dz
K 11 ij
¼
K 12 ij ¼ K 13 ij
¼
K 14 ij ¼
Z Z Z
0 l 0 l 0 l 0
0
K 55 ij
0
K 22 ij
¼
K 23 ij ¼ K 24 ij ¼
Z Z Z
K 56 ij
0
K 57 ij
l
E66 w0i w0j dz 0 l
E67 w0i w0j dz
K 66 ij
0 l
ðE56 þ E68 Þw0i w0j dz
K 67 ij
0
K 25 ij
Z lh i ¼ E26 w0i w0j þ E66 w0i wj dz
K 26 ij
Z lh i ¼ E36 w0i w0j þ E67 w0i wj dz
0
K 77 ij
0
K 27 ij ¼ K 33 ij
¼
K 34 ij ¼
Z lh i E46 w0i w0j þ ðE68 E56 Þw0i wj dz Z Z
0
219
Z lh i ¼ E47 w0i w0j þ ðE78 E57 Þw0i wj dz 0 Z l ¼ ðE55 þ 2E58 þ E88 Þw0i w0j dz 0 Z lh i ¼ ðE25 þ E28 Þw0i w0j þ ðE56 þ E68 Þw0i wj dz 0 Z lh i ¼ ðE35 þ E38 Þw0i w0j þ ðE57 þ E78 Þw0i wj dz 0 Z lh i ¼ ðE45 þ E48 Þw0i w0j þ ðE88 E55 Þw0i wj dz 0 Z lh i ¼ E22 w0i w0j þ E26 w0i wj þ wi w0j þ E66 wi wj dz 0 Z lh i ¼ E23 w0i w0j þ E27 w0i wj þ E36 wi w0j þ E67 wi wj dz 0 Z lh ¼ E24 w0i w0j þ ðE28 E25 Þw0i wj þ E46 wi w0j 0 i þ ðE68 E56 Þwi wj dz Z lh i ¼ E33 w0i w0j þ E37 w0i wj þ wi w0j þ E77 wi wj dz 0 Z lh ¼ E34 w0i w0j þ ðE38 E35 Þw0i wj þ E47 wi w0j 0 i þ ðE78 E57 Þwi wj dz Z lh ¼ E44 w0i w0j þ ðE48 E45 Þ w0i wj þ wi w0j 0 i þ ðE88 2E58 þ E55 Þwi wj dz ð43Þ
In Eq. (42), the force vector is given by Z l fi3 ¼ qwi dz
l
E77 w0i w0j dz 0
ð44Þ
0
l
ðE57 þ E78 Þw0i w0j dz
This completes the finite element formulation of the present theory.
0
K 35 ij
Z lh i ¼ E27 w0i w0j þ E67 w0i wj dz
K 36 ij
Z lh i ¼ E37 w0i w0j þ E77 w0i wj dz
0
8. Numerical results and discussion
0
For the verification purpose, the results by present approach are compared with the previous results [6,8] Table 1 Deflections of a simply-supported I-section beam under uniformly distributed load Stacking sequence
ABAQUS
Ref. [6]
[0]16 [15/15]4s [30/30]4s [45/45]4s [60/60]4s [75/75]4s [0/90]4s
6.340 6.989 9.360 13.479 17.023 18.490 9.400
6.283 6.941 9.322 13.449 16.993 18.453 9.349
Ref. [8]
Present
s = 0
rs = 0
s = 0
rs = 0
6.103 6.610 8.281 11.340 15.119 17.643 9.153
6.233 6.899 9.290 13.421 16.962 18.411 9.299
6.129 6.637 8.307 11.363 15.151 17.683 9.189
6.259 6.923 9.314 13.446 16.992 18.449 9.381
220
J. Lee / Composite Structures 70 (2005) 212–222 10cm
9.0
0.5cm orthotropy (no shear)
8.0
FEM (no shear) orthotropy (shear)
7.0
FEM (shear)
6.0
0.5cm
20cm
5.0 v 4.0 3.0 2.0
Fig. 3. Thin-walled I-section composite beam.
1.0
and ABAQUS [10]. In ABAQUS analysis, S9R5 shell elements are used. A simply-supported I-section beam with 2.5 m length under uniformly distributed load of 1 kN/m with dimension of (50 · 50 · 2.08) mm is considered. The following engineering constants are used. E1 ¼ 53:78 GPa;
E2 ¼ E3 ¼ 17:93 GPa
G23 ¼ 3:45 GPa; G12 ¼ G13 ¼ 8:96 GPa m23 ¼ 0:34; m12 ¼ m13 ¼ 0:25
ð45Þ
The maximum vertical deflections based on two different assumptions (rs = 0 and s = 0) are compared with the previous results by Barbero [6], results by the author using classical beam model [8], and ABAQUS [10] in Table 1 for several stacking sequences. All the results are in good agreement. Generally, the results with
0.0 0
15
30
45 fiber angle
60
75
90
Fig. 5. Central deflection with respect to the fiber angle of a simplysupported anti-symmetric angle-ply I-section composite beam with l/b3 = 20 under uniformly-distributed load.
plane stress assumption (rs = 0) show excellent agreement with ABAQUS solution. For all the stacking sequences considered, the present approach exhibits better results than that of classical beam assumption. As a next example, a thin-walled I-section composite beam is considered in order to investigate the coupling and transverse shear deformation effects. The geometry of the section of the beam is shown in Fig. 3, and the following material properties are considered in this investigation: E1 =E2 ¼ 25 G12 =E2 ¼ 0:5 G13 =E2 ¼ 0:6
9.0 orthotropy (no shear)
8.0
m12 ¼ 0:25
FEM (no shear) orthotropy (shear)
7.0
ð46Þ
For convenience, the following nondimensional value of vertical displacements are used:
FEM (shear)
6.0
v ¼
5.0 v
vE2 b33 ql4
ð47Þ
4.0
Table 2 Ratio of coupling stiffnesses with respect to the bending stiffness when the top and bottom flanges, and the web are all antisymmetric angleply
3.0 2.0 1.0 0.0
0
15
30
45 fiber angle
60
75
90
Fig. 4. Central deflection with respect to the fiber angle of a simplysupported anti-symmetric angle-ply I-section composite beam with l/b3 = 50 under uniformly-distributed load.
Fiber angle
E15/E33
E27/E33
E38/E33
0 15 30 45 60 75 90
0.000 0.023 0.042 0.042 0.022 0.007 0.000
0.000 0.023 0.042 0.042 0.022 0.007 0.000
0.000 0.002 0.004 0.004 0.002 0.001 0.000
J. Lee / Composite Structures 70 (2005) 212–222
221
0.6
The beam is assumed to be under uniformly-distributed loading with simply-supported boundary condition. The top and bottom flanges and the web are considered as antisymmetric angle-ply laminates [h/h]. For all the analyses, the assumption rs = 0 is made. Variation of the deflection at the center of the beam with respect to the fiber angle change is shown in Figs. 4 and 5 for l/b3 = 50 and l/b3 = 20. In generating Figs. 3 and 4, the finite element solution with no shear is calculated based on previous paper [7]. The closed-form solution of the maximum deflection for the uncoupled equations (Eq. (39)) can be directly calculated for the beam with uniformly-distributed loading and simply-supported boundary conditions as [11]:
0.5
vmax ¼
Table 3 Ratio of coupling stiffnesses with respect to the bending stiffness when the top flange and web is unidirectional while the bottom flange is symmetric angle-ply Fiber angle
E13/E33
E24/E33
E68/E33
0 15 30 45 60 75 90
0.000 0.894 3.751 5.196 5.537 5.611 0.000
0.000 0.001 0.003 0.004 0.005 0.005 0.000
0.000 0.274 0.804 0.789 0.634 0.544 0.000
0.4
v 0.3
0.2 orthotropy (no shear) FEM (no shear)
0.1
orthotropy (shear) FEM (shear)
0.0
0
15
30
45 fiber angle
60
75
90
Fig. 6. Central deflection with respect to the fiber angle of a simplysupported I-section composite beam with symmetric angle-ply for the bottom flange and l/b3 = 50 under uniformly-distributed load.
0.6
0.5
0.4
v 0.3
0.2 orthotropy (no shear) FEM (no shear)
0.1
orthotropy (shear) FEM (shear)
0.0
0
15
30
45 fiber angle
60
75
90
Fig. 7. Central deflection with respect to the fiber angle of a simplysupported I-section composite beam with symmetric angle-ply for the bottom flange and l/b3 = 20 under uniformly-distributed load.
5ql4 ql2 þ 384ðEI x Þcom 8ðGAx Þcom
ð48Þ
In Eq. (48), the first term denotes the deflection by classical beam theory, and the second term is the deflection by shear deformation. For this stacking sequence, the coupling stiffnesses E15, E27 and E38 do not vanish while all the other coupling stiffnesses become zero. That is, the orthotropy solution given in Eq. (48) might not be accurate. However, the coupling stiffnesses are very small compared to bending stiffness E33 (Table 2). Consequently, the coupling effects coming from the material anisotropy become negligible for two different spanto-height ratios. It also shows that the transverse shear effects are also negligbly small even for the beam with the lower span-to-height ratio (l/b3 = 20) indicating that the simple orthotropy solution of the classical beam theory is sufficiently accurate for these cases. To investigate the coupling and transverse shear effects further, the same configuration with the previous example except the laminate stacking sequence is considered. The stacking sequence of the top flange and the web of the beam is unidirectional while the bottom flange is assumed to be [h/h/h/h]. For this stacking sequence, the coupling stiffnesses E13, E24 and E68 do not vanish while all the other coupling stiffnesses become zero. Especially, E13 becomes no more negligibly small as given in Table 3. For l/b3 = 50, as fiber angle increases, the orthotropic and finite element solutions show discrepancy indicating the coupling effects become significant (Fig. 6). The transverse shear little affects the behavior of the beam. For lower span-to-height ratio (Fig. 7), however, the solutions excluding shear effects remarkably underestimate the deflection for all the range of fiber angle. Furthermore, the orthotropic solutions disagree with the finite element solutions as orthotropy of the beam gets higher.
222
J. Lee / Composite Structures 70 (2005) 212–222
9. Concluding remarks
References
An analytical model was developed to study the flexural behavier of a laminated composite beam with an I-section. The model is capable of predicting accurate deflection for various configuration including boundary conditions, laminate orientation and span-to-height ratio. To formulate the problem, a one-dimensional displacement-based finite element method is employed. The assumption that normal stress in contour direction vanishes (rs = 0) seems more appropriate than free strain assumption in contour direction (s = 0). The shear effects become significant for lower span-to-height ratio and higher degrees of orthotropy of the beam. The orthotropy solution is accurate for lower degrees of material anisotropy, but, becomes inappropriate as the anisotropy of the beam gets higher, and fully coupled equations should be considered for accurate analysis of thin-walled composite beams. The presented analytical model is found to be appropriate and efficient in analyzing flexural problem of thin-walled laminated composite beams.
[1] Vlasov VZ. Thin-walled elastic beams. second ed. Jerusalem, Israel: Israel Program for Scientific Translation; 1961. [2] Gjelsvik A. The theory of thin-walled bars. New York: John Wiley and Sons Inc.; 1981. [3] Bauld NR, Tzeng LS. A Vlasov theory for fiber-reinforced beams with thin-walled open cross section. Int J Solids Struct 1984;20(3):277–97. [4] Davalos JF, Salim HA, Qiao P, Lopez-Anido R. Analysis and design of pultruded FRP shapes under bending. Compos: Part B 1996;27B:295–305. [5] Ascione L, Feo L, Mancusi G. On the statical behavior of fiberreinforced polymer thin-walled beams. Compos: Part B 2000;31B: 643–654. [6] Barbero EJ. Introduction to composite materials design. Taylor & Francis; 1999. [7] Qin Z, Librescu L. On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations. Compos Struct 2002;56(2):345–58. [8] Lee J, Lee S. Flexural–torsional behavior of thin-walled composite beams. Thin-walled Struct 2004;42(9):1293–305. [9] Jones RM. Mechanics of composite materials. Hemishere Publishing Corporation; 1975. [10] ABAQUS/Standard UserÕs Manual, Version 6.4, Hibbit, Kalsson & Sorensen Inc.; 2003. [11] Reddy JN. Mechanics of laminated composite plates: theory and analysis. CRC Press; 1997.
Acknowledgment The support of the research reported here by Korea Research Foundation through Grant KRF-2003-041D00595 is gratefully acknowledged.
Flexural analysis of thin-walled composite beams using shear-deformable beam theory Jaehong Lee
*
Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Korea Available online 25 September 2004
Abstract This paper presents a flexural analysis of I-shaped laminated composite beams. A general analytical model applicable to thinwalled I-section composite beams subjected to vertical load is developed. This model is based on the shear-deformable beam theory, and accounts for the flexural response of the thin-walled composites for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric. Governing equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for thin-walled composites under vertical loading, addressing the effects of fiber angle and span-to-height ratio of the composite beam. 2004 Elsevier Ltd. All rights reserved. Keywords: Thin-walled structures; Laminated composites; Flexural response; 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 stiffness and strength to weight. In the construction industry, recent applications have shown the structural and cost efficiency of FRP structural shapes, such as thin-walled open sections through pultrusion manufacturing process. Thin-walled open section members made of isotropic materials have been studied by many researchers [1,2]. Bauld and Tzeng [3] extended VlasovÕs thin-walled bar theory [1] to symmetric fiber-reinforced laminates to develop the linear and nonlinear theories for the bending and twisting of thin-walled composite beams. Davalos et al. [4] studied the bending response of pultruded composite beams with various I and box sections experimentally and analytically. They found that the micromechanics formulae can be used to predict the ply stiffnesses of the laminate. *
Tel.: +82 2 3408 3287; fax: +82 2 3408 3331. E-mail address: [email protected]
0263-8223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.08.023
Ascione et al. [5] presented the statical behavior of fiber-reinforced polymer thin-walled beams taking into account the effects of shear deformation. They compare their results with VlasovÕs classical solution. Barbero [6] investigated the behavior of thin-walled composite beams in his monograph. Recently, Qin and Librescu [7] proposed shear-deformable analytical model for anisotropic thin-walled beams including nonclassical effects. More recently, Lee and Lee [8] presented analytical model which accounts for flexural–torsional behavior of I-section composite beams. They developed one-dimensional finite element model to investigate the flexural–torsional behavior of thin-walled composite beams. In this paper, the analytical model developed in [8] is extended by incorporating transverse shear deformation. This model is based on the first-order sheardeformable beam theory, and accounts for all the structural coupling coming from the material anisotropy. Warping effects are also included. Governing equations are derived from the principle of the stationary value of total potential energy. Numerical results
J. Lee / Composite Structures 70 (2005) 212–222
are obtained for thin-walled composites under vertical loading, addressing the effects of fiber angle and spanto-height ratio of the composite beam.
213
the midsurface shear strains in the contour can be expressed with respect to the transverse shear and warping shear strains. cnz ðs; zÞ ¼ cxz ðzÞ sin hðsÞ cyz ðzÞ cos hðsÞ þ cx ðzÞqðsÞ ð2aÞ
2. Kinematics This paper requires three sets of coordinate systems: an orthogonal Cartesian coordinate system (x, y, z), an orthogonal coordinate system (n, s, z), and a contour coordinate s along the profile of the section with its origin at any point O on the profile section. Three sets of coordinate systems are mutually interrelated and shown in Fig. 1. The n axis as shown in Fig. 1 is normal to the middle surface of a plate element, the s axis is tangent to the middle surface and is directed along the contour line of the cross section. The basic assumptions regarding the kinematics of thin-walled composites are state as follows: 1. The contour of the thin wall does not deform in its own plane. 2. Transverse shear strains cxz , cyz and warping shear cx are incorporated. It is assumed that they are uniform over the cross-sections. According to assumption 1, the midsurface displacement components u, v at a point A in the contour coordinate system can be expressed in terms of a displacements U, V of the pole P in the x, y directions, respectively, and the rotation angle U about the pole axis,
csz ðs; zÞ ¼ cxz ðzÞ cos hðsÞ þ cyz ðzÞ sin hðsÞ þ cx ðzÞrðsÞ ð2bÞ Further, it is assumed that midsurface shear strain in s n direction is zero (csn ¼ 0). From the definition of the shear strain, csz can also be given for each element of middle surface as: csz ðs; zÞ ¼
o w ov þ os oz
ð3Þ
After substituting for v from Eq. (1) and considering the following geometric relations, dx ¼ ds cos h
ð4Þ
dy ¼ ds sin h
ð5Þ
Eq. (3) can be integrated with respect to s from the origin to an arbitrary point on the contour. By compar ing Eqs. (2) and (3), the midsurface axial displacement w can then be obtained as ðs; zÞ ¼ W ðzÞ þ Wy ðzÞxðsÞ þ Wx ðzÞyðsÞ þ Wx ðzÞxðsÞ w ð6Þ where Wx, Wy and Wx represent rotations of the cross section with respect to x, y and x, respectively, given by:
uðs; zÞ ¼ U ðzÞ sin hðsÞ V ðzÞ cos hðsÞ UðzÞqðsÞ
ð1aÞ
Wy ¼ cxz U 0
ð7Þ
vðs; zÞ ¼ U ðzÞ cos hðsÞ þ V ðzÞ sin hðsÞ þ UðzÞrðsÞ
ð1bÞ
Wx ¼ cyz V 0
ð8Þ
Wx ¼ cx U0
ð9Þ
These equations apply to the whole contour. The out can now be found from of-plane shell displacement w the assumption 2. For each element of middle surface,
y
s n
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 assuming the first order variation of inplane displacements v, w through the thickness of the contour as:
P r O
q Θ
z
When the transverse shear effect is ignored, Eq. (9) degenerate to Wy = U 0 , Wx = V 0 and Wx = U 0 , and as a result, the number of unknown variables reduces to four leading to the Euler–Bernoulli beam model. The prime ( 0 ) is used to indicate differentiation with respect to z; and x is the so-called sectorial coordinate or warping function given by Z xðsÞ ¼ rðsÞ ds ð10Þ
uðs; z; nÞ ¼ uðs; zÞ
ð11aÞ
ðs; zÞ vðs; z; nÞ ¼ vðs; zÞ þ nw s
ð11bÞ
x
Fig. 1. Definition of coordinates in thin-walled open section.
214
J. Lee / Composite Structures 70 (2005) 212–222
ðs; zÞ ðs; zÞ þ nw wðs; z; nÞ ¼ w ð11cÞ z and w denote the rotations of a transverse where, w s z normal about the z and s axis, respectively. These functions can be determined by considering that the midsurface shear strains cnz is given by definition: cnz ðs; zÞ ¼
o w o u þ on oz
ð12Þ
can By comparing Eqs. (2) and (12), the function w z be written as ¼ Wy sin h Wx cos h Wx q w ð13Þ z Similarly, using the assumption that the shear strain can be csn should vanish at midsurface, the function w s obtained u ¼ o w s os
respect to the shear center, and twisting curvature in the beam, respectively defined as z ¼ W 0
ð18aÞ
jx ¼ W0x
ð18bÞ
jy ¼ W0y
ð18cÞ
jx ¼ W0x
ð18dÞ
jsz ¼ U0 Wx
ð18eÞ
The resulting strains can now be obtained with respect to the beam strain components from Eqs. (15) and (17) as z ¼ z þ ðx þ n sin hÞjy þ ðy n cos hÞjx þ ðx nqÞjx
ð14Þ
ð19aÞ csz ¼ cxz cos h þ cyz sin h þ cx r þ njsz
ð19bÞ
3. Strains
cnz ¼ cxz sin h cyz cos h þ cx q
ð19cÞ
The strains associated with the small-displacement theory of elasticity are given by dividing the midsurface strains and curvatures of the contour:
4. Variational formulation
s ðs; z; nÞ ¼ s ðs; zÞ þ n js ðs; zÞ
ð15aÞ
jz ðs; zÞ z ðs; z; nÞ ¼ z ðs; zÞ þ n
ð15bÞ
csz ðs; z; nÞ ¼ csz ðs; zÞ þ n jsz ðs; zÞ
ð15cÞ
cnz ðs; z; nÞ ¼ cnz ðs; zÞ þ n jnz ðs; zÞ
ð15dÞ
Total potential energy of the system is calculated by sum of strain energy and potential energy, P¼UþV
where s ¼
ov ; os
s ¼ j
ow s os
ð16aÞ
z ¼
o w ; oz
z ¼ j
ow z oz
ð16bÞ
sz ¼ j
ow ow z þ s; os oz
nz ¼ 0 j
ð16cÞ
All the other strains are identically zero. In Eq. (16), s and j s are assumed to be zero, and z , j z and j sz are midsurface axial strain and biaxial curvatures of the shell, respectively. The above shell strains can be converted to beam strain components by substituting Eqs. (1) and (11) into Eq. (16) z ¼ z þ xjy þ yjx þ xjx
ð17aÞ
z ¼ jy sin h jx cos h jx q j
ð17bÞ
sz ¼ jsz j
ð17cÞ
where z , jx, jy, jx and jsz are axial strain, biaxial curvatures in the x and y direction, warping curvature with
where U is the strain energy Z 1 U¼ ðrz z þ rsz csz þ rnz cnz Þ dv; 2 v
ð20Þ
ð21Þ
The strain energy is calculated by substituting Eq. (17) into Eq. (21) Z n 1 U¼ rz ½z þ ðx þ n sin hÞjy þ ðy n cos hÞjx 2 v þðx nqÞjx þ rsz ½cxz cos h þ cyz sin h þ cx r þ njsz o ð22Þ þrnz ½cxz sin h cyz cos h þ cx q dv The variation of strain energy, Eq. (22), can be stated as Z ln N z dz þ M y djy þ M x djx þ M x djx þ V x dcxz 0 o þV y dcyz þ T dcx þ M t djsz dz ð23Þ
dU ¼
where Nz, Mx, My, Mx, Vx, Vy, T and Mt are axial force, bending moments in the x and y directions, warping moment (bimoment), shear force in the x and y direction, and torsional moments, respectively, defined by integrating over the cross-sectional area A as: Z Nz ¼ rz ds dn ð24aÞ A
J. Lee / Composite Structures 70 (2005) 212–222
Z
My ¼
rz ðx þ n sin HÞ ds dn
ð24bÞ
Mx ¼
rz ðy n cos HÞ ds dn
ð29aÞ
ð29bÞ
ð24dÞ
¼ Q 16 Q12 Q26 Q 16 22 Q
ð24eÞ
2 66 Q26 ¼ Q Q 66 22 Q
ð29cÞ
ð24cÞ
A
Z
Mx ¼
rz ðx nqÞ ds dn
A
Vx ¼
Z
ðrsz cos h þ rnz sin hÞ ds dn A
Vy ¼
Z
and for free strain assumption, ðrsz sin h rnz cos hÞ ds dn
ð24fÞ
A
T ¼
Z
ðrsz r rnz qÞ ds dn
ð24gÞ
A
Mt ¼
for free stress assumption, 2 11 Q12 ¼ Q Q 11 22 Q
A
Z
215
Z
ð24hÞ
rsz n ds dn A
The variation of the work done by external force can be stated as Z l dV ¼ qdV dz ð25Þ 0
where q is transverse load. Using the principle which the variation of the total potential energy is zero, the following weak statement is obtained: Z ln 0¼ N z dW 0 þ M y dW0y þ M x dW0x þ M x dW0x 0
þV x dðU 0 þ Wy Þ þ V y dðV 0 þ Wx Þ þ T dðU0 þ Wx Þ o þM t dðU0 Wx Þ þ qdV dz ð26Þ
5. Constitutive equations of plate elements
11 ¼ Q Q 11
ð30aÞ
¼ Q 16 Q 16
ð30bÞ
66 ¼ Q Q 66
ð30cÞ
The constitutive relation for out-of-plane stress and strain is given by 55 cnz rnz ¼ Q ð31Þ In Eq. (24), the stress resultants Nz, Mx, My, Mx, Mx, Mt, Vx, Vy and T can now be expressed with respect to the generalized strains (z ; jy ; jx ; jx ; jsz ; cxz ; cyz ; cx ) by combining Eqs. (24), (28) and (19). Consequently, the constitutive equations for a thin-walled laminated composite are obtained as 38 9 9 2 8 E11 E12 E13 E14 E15 E16 E17 E18 > z > Nz > > > > > > > 7> > > 6 >j > >M > > > > E E E E E E E 7 6 > > > > y > y 22 23 24 25 26 27 28 > > > 7 6 > > > > > > > > > 7 6 > > > jx > Mx > E33 E34 E35 E36 E37 E38 7> > > > > > > 6 > > > > > > 7> = = 6 <
The constitutive equation for in-plane stresses and strains in an arbitrary z–s coordinate system is then written 8 9k 2 11 Q < rz = 4 12 r ¼ Q : s; 16 rsz Q
12 Q 22 Q 26 Q
8 9 16 3k < z = Q 26 5 s Q 66 : csz ; Q
ð27Þ
ij are transformed reduced stiffnesses [9] and are where Q made up of material property with respect to the each layer. The above constitutive equation can be simplified by using free stress in contour direction (rs = 0) or free strain in contour direction (s = 0) assumption as
rz rsz
"
k ¼
Q 11 Q 16
Q 16 Q 66
#k
z csz
ð28Þ
In Eq. (32), Eij are stiffnesses of the thin-walled composite. It appears that the laminate stiffnesses Eij depend on the cross section of the composites given by Z E11 ¼ A11 ds Zs E12 ¼ ðA11 x þ B11 sin hÞ ds Zs E13 ¼ ðA11 y B11 cos hÞ ds Zs E14 ¼ ðA11 x B11 qÞ ds Zs B16 ds E15 ¼ Zs A16 cos h ds E16 ¼ s
216
E17 ¼ E18 ¼ E22 ¼ E23 ¼
J. Lee / Composite Structures 70 (2005) 212–222
Z
E68 ¼
A16 sin h ds Zs
E77 ¼
A16 r ds Zs Zs
E24 ¼
2
2
ðA11 x þ 2B11 x sin h þ D11 sin hÞ ds ðA11 xy þ B11 ðy sin h x cos hÞ D11 sin h cos hÞ ds
E26 ¼ E27 ¼ E28 ¼ E33 ¼
Z
Zs Zs Zs Zs
ðA11 xx þ B11 ðx sin h qxÞ D11 q sin hÞ ds ðB16 x þ D16 sin hÞ ds ðA16 x þ B16 sin hÞ cos h ds ðA16 x þ B16 sin hÞ sin h ds
E35 ¼ E36 ¼ E37 ¼
Z
Zs Zs Zs
E78 ¼ E88 ¼
E44 ¼ E45 ¼ E46 ¼ E47 ¼ E48 ¼
Z
Zs Zs Zs Zs Zs
E56 ¼ E57 ¼ E58 ¼ E66 ¼
ðA66 r sin h þ A55 q cos hÞ ds ðA66 r2 þ A55 q2 Þ ds
ðA16 y B16 cos hÞ sin h ds ðA16 y B16 cos hÞr ds
The above expressions are general for arbitrary cross sections. For I-section shapes, the explicit expressions become: ¼ Ak11 bk ¼ B311 b3 ¼ Aa11 ba y a Ba11 ba ¼ Bk16 bk ¼ Aa16 ba ¼ A316 b3 ¼ Aa16 y a ba
bað3Þ þ D311 b3 12 bð3Þ E24 ¼ ½Aa11 y a Ba11 a 12 E25 ¼ D316 b3 E27 ¼ B316 b3 ð3Þ
ðA11 x2 2B11 xq þ D11 q2 Þ ds
a a E33 ¼ ½Aa11 y ð2Þ a 2B11 y a þ D11 ba þ
ðB16 x D16 qÞ ds
E34 ¼
ðA16 x B16 qÞ cos h ds ðA16 x B16 qÞ sin h ds ðA16 x B16 qÞr ds
Zs B66 cos h ds Zs B66 sin h ds Zs B66 r ds ðA66 cos2 h þ A55 sin2 hÞ ds
E35 E36 E37 E38
b3 3 B 12 11 a ¼ B16 y a ba Da16 ba ¼ ðAa16 y a Ba16 Þba ¼ A316 y 3 b3 ¼ ðAa16 y a Ba16 Þy a ba
s
ðA66 A55 Þ sin h cos h ds
ð35Þ
ð3Þ
a a E44 ¼ ½Aa11 y ð2Þ a 2B11 y a þ D11
E47 E55 E56 E57 E58 E66 E68 E77
bað3Þ b3 3 þ D 12 12 11
¼ B316 b3 y 3 ¼ Dk66 bk ¼ Ba66 ba ¼ B366 b3 ¼ Ba66 y a ba ¼ Aa66 ba þ A355 b3 ¼ Aa66 y a ba þ A344 y 3 b3 ¼ Aa55 ba þ A366 b3 ð2Þ
s
E67 ¼
b3 3 A 12 11
ð3Þ
D66 ds
Z
ð33Þ
where, Aij, Bij and Dij matrices are extensional, coupling and bending stiffness, respectively, defined by Z ij ð1; n; n2 Þ dn ðAij ; Bij ; Dij Þ ¼ Q ð34Þ
E22 ¼ Aa11
Z
Zs
Zs
ðA16 y B16 cos hÞ cos h ds
ðA11 yx B11 ðx cos h þ qyÞ þ D11 q cos hÞ ds
s
E55 ¼
Z
ðB16 y D16 cos hÞ ds
ðA11 y 2 2B11 y cos h þ D11 cos2 hÞ ds
s
E38 ¼
ðA66 sin2 h þ A55 cos2 hÞ ds
E11 E12 E13 E15 E16 E17 E18
ðA16 x þ B16 sin hÞr ds
s
E34 ¼
ðA66 r cos h A55 q sin hÞ ds
s
s
E25 ¼
Zs s
s
Z
Z
E88 ¼
Aa66 y að2Þ ba
Aa bð3Þ b ð2Þ þ 55 a þ A355 b3 y 3 þ 3 12 12
!
J. Lee / Composite Structures 70 (2005) 212–222
dWx : M x
By substituting Eqs. (32) and (18) into Eq. (36), the explicit form of the governing equations yield: E11 W 00 þ E16 U 00 þ E17 V 00 þ ðE15 þ E18 ÞU00 þ E12 W00y þ E16 W0y
y1
þ E13 W00x þ E17 W0x þ E14 W00x þ ðE18 E15 ÞW0x ¼ 0 x h3
b3
þ E36 W00x þ E67 W0x þ E46 W00x þ ðE68 E56 ÞW0x ¼ 0
y2
þ E37 W00x þ E77 W0x þ E47 W00x þ ðE78 E57 ÞW0x ¼ q
ð38cÞ
ðE15 þ E18 ÞW 00 þ ðE56 þ E68 ÞU 00 þ ðE57 þ E78 ÞV 00
Fig. 2. Geometry of a thin-walled composite.
where the superscript in the parenthesis ( ) denotes the power, and repeated index denotes summation. Index k varies from 1 to 3 whereas a varies from 1 to 2 implying top and bottom flanges (1, 2) and web (3) as shown in Fig. 2. bk denotes width of flanges and web. All the other Eij are zero.
þ ðE55 þ 2E58 þ E88 ÞU00 þ ðE25 þ E28 ÞW00y þ ðE56 þ E68 ÞW0y þ ðE56 þ E68 ÞW00x þ ðE57 þ E78 ÞW0x þ ðE45 þ E48 ÞW00x þ ðE88 E55 ÞW0x ¼ 0
ð38dÞ
E12 W 00 E16 W 0 þ E26 U 00 E66 U 0 þ E27 V 00 E67 V 0 þ ðE25 þ E28 ÞU00 ðE56 þ E68 ÞU0 þ E22 W00y E66 Wy þ E23 W00x þ ðE27 E36 ÞW0x E67 Wx þ E24 W00x
6. Governing equations
þ ðE28 E25 E46 ÞW0x þ ðE56 E68 ÞWx ¼ 0
The governing equations of the present approach can be derived by integrating the derivatives of the varied quantities by parts and collecting the coefficients of dW, dU, dV, dU, dWy, dWx and dWx: ð36aÞ N 0z ¼ 0 0 x
ð36bÞ
0 y
ð36cÞ 0
ð38bÞ
E17 W 00 þ E67 U 00 þ E77 V 00 þ ðE57 þ E78 ÞU00 þ E27 W00y þ E67 W0y b2
V ¼q
ð38aÞ
E16 W 00 þ E66 U 00 þ E67 V 00 þ ðE56 þ E68 ÞU00 þ E26 W00y þ E66 W0y
h2
V ¼0
ð37gÞ
y b1 h1
z
217
ðM t þ T Þ ¼ t
ð36dÞ
M 0y V x ¼ 0
ð36eÞ
M 0x V y ¼ 0
ð36fÞ
M 0x þ M t T ¼ 0
ð36gÞ
The natural boundary conditions are of the form: dW : N z
ð37aÞ
dU : V x
ð37bÞ
dV : V y
ð37cÞ
dU : T þ M t
ð37dÞ
dWy : M y
ð37eÞ
dWx : M x
ð37fÞ
ð38eÞ
E13 W 00 E17 W 0 þ E36 U 00 E67 U 0 þ E37 V 00 E77 V 0 þ ðE35 þ E38 ÞU00 ðE57 þ E78 ÞU0 þ E23 W00y þ ðE36 E27 ÞW0y E67 Wy þ E33 W00x E77 Wx þ E34 W00x þ ðE38 E35 E47 ÞW0x þ ðE57 E78 ÞWx ¼ 0
ð38fÞ
E14 W 00 þ ðE15 E18 ÞW 0 þ E46 U 00 þ ðE56 E68 ÞU 0 þ E47 V 00 þ ðE57 E78 ÞV 0 þ ðE45 þ E48 ÞU00 þ ðE55 E88 ÞU0 þ E24 W00y þ ðE46 þ E25 E28 ÞW0y þ ðE56 E68 ÞWy þ E34 W00x þ ðE47 þ E35 E38 ÞW0x þ ðE57 E78 ÞWx þ E44 W00x ðE88 2E58 þ E55 ÞWx ¼ 0
ð38gÞ
Eq. (38) are most general form for a thin-walled laminated composite beam. For general anisotropic materials, the dependent variables, U, V, W, U, Wx, Wy and Wx are fully-coupled implying that the beam undergoes a coupled behavior involving bending, twising, extension, transverse shearing, and warping. If all the coupling effects are neglected, Eq. (38) can be simplified to the uncoupled differential equations as: ðEAÞcom W 00 ¼ 0
ð39aÞ
ðGAy Þcom ðU 00 þ W0y Þ ¼ 0
ð39bÞ
ðGAx Þcom ðV 00 þ W0x Þ ¼ q
ð39cÞ
218
J. Lee / Composite Structures 70 (2005) 212–222
ðGJ 1 Þcom U00 ðGJ 2 Þcom W0x ¼ 0
ð39dÞ
ðEI y Þcom W00y ðGAy Þcom ðU 0 þ Wy Þ ¼ 0
ð39eÞ
ðEI x Þcom W00x ðGAx Þcom ðV 0 þ Wx Þ ¼ 0
ð39fÞ
ðEI x Þcom W00x þ ðGJ 2 Þcom U0 ðGJ 1 Þcom Wx ¼ 0
ð39gÞ
From above equations, (EA)com represents axial rigidity, (GAx)com, (GAy)com represent shear rigidities with respect to x and y axis, (EIx)com and (EIy)com represent flexural rigidities with respect to x and y axis, (EIx)com represents warping rigidity, and (GJ1)com, (GJ2)com represent torsional rigidities of the thin-walled composite, respectively, written as ðEAÞcom ¼ E11
ð40aÞ
ðEI y Þcom ¼ E22
ð40bÞ
ðEI x Þcom ¼ E33
ð40cÞ
ðEI x Þcom ¼ E44
ð40dÞ
ðGAy Þcom ¼ E66
ð40eÞ
ðGAx Þcom ¼ E77
ð40fÞ
ðGJ 1 Þcom ¼ E55 þ E88
ð40gÞ
ðGJ 2 Þcom ¼ E55 E88
ð40hÞ
For bending analysis with respect to x-axis, only Eqs. (39c) and (39f) are involved, and these equations are well-known Timoshenko beam equations.
7. Finite element model The present theory for thin-walled composite beams described in the previous section was implemented via a displacement based one-dimensional finite element method. The generalized displacements are expressed over each element as a linear combination of the onedimensional Lagrange interpolation function wj associated with node j and the nodal values: n X
W ¼
wj wj ;
U¼
j¼1
U¼
n X
uj wj ;
V ¼
j¼1
/j wj ;
j¼1
Wx ¼
n X
n X j¼1
Wy ¼
n X
Wx ¼
vj w j ;
j¼1
wyj wj ;
j¼1
wxj wj ;
n X
n X j¼1
wxj wj
ð41Þ
Substituting these expressions into the weak statement in Eq. (26), the finite element model of a typical element can be expressed as 38 9 8 9 2 K 11 K 12 K 13 K 14 K 15 K 16 K 17 > > > > >0> >w> > 7> 6 > > > > > > > > > > 7 6 > > > > 0 u K K K K K K > > > 22 23 24 25 26 27 7> 6 > > > > > > > > 7> 6 > > > > > > > > > > > 7 6 f v K K K K K > > > > 33 34 35 36 37 3 > > > 7> 6 7< = < = 6 7 6 ¼ / K K K K 0 44 45 46 47 7 6 > > > > > > > 7> 6 > > > > > > > 7> 6 w 0 K K K > > > > 55 56 57 y > > > 7> 6 > > > > > > > 7> 6 > > > > > > > > 7 6 > > > > 0 K K w 66 67 5> x > > 4 > > > > > > > > ; : ; : > 0 sym: K 77 wx ð42Þ where [K] is the element stiffness matrix given by
J. Lee / Composite Structures 70 (2005) 212–222
Z
l
E11 w0i w0j dz
K 37 ij
E17 w0i w0j dz
K 44 ij
E18 w0i w0j dz
K 45 ij
ðE15 þ E18 Þw0i w0j dz
K 46 ij
K 15 ij
Z lh i ¼ E12 w0i w0j þ E16 w0i wj dz
K 47 ij
K 16 ij
Z lh i ¼ E13 w0i w0j þ E17 w0i wj dz
K 17 ij
Z lh i ¼ E14 w0i w0j þ ðE18 E15 Þw0i wj dz
K 11 ij
¼
K 12 ij ¼ K 13 ij
¼
K 14 ij ¼
Z Z Z
0 l 0 l 0 l 0
0
K 55 ij
0
K 22 ij
¼
K 23 ij ¼ K 24 ij ¼
Z Z Z
K 56 ij
0
K 57 ij
l
E66 w0i w0j dz 0 l
E67 w0i w0j dz
K 66 ij
0 l
ðE56 þ E68 Þw0i w0j dz
K 67 ij
0
K 25 ij
Z lh i ¼ E26 w0i w0j þ E66 w0i wj dz
K 26 ij
Z lh i ¼ E36 w0i w0j þ E67 w0i wj dz
0
K 77 ij
0
K 27 ij ¼ K 33 ij
¼
K 34 ij ¼
Z lh i E46 w0i w0j þ ðE68 E56 Þw0i wj dz Z Z
0
219
Z lh i ¼ E47 w0i w0j þ ðE78 E57 Þw0i wj dz 0 Z l ¼ ðE55 þ 2E58 þ E88 Þw0i w0j dz 0 Z lh i ¼ ðE25 þ E28 Þw0i w0j þ ðE56 þ E68 Þw0i wj dz 0 Z lh i ¼ ðE35 þ E38 Þw0i w0j þ ðE57 þ E78 Þw0i wj dz 0 Z lh i ¼ ðE45 þ E48 Þw0i w0j þ ðE88 E55 Þw0i wj dz 0 Z lh i ¼ E22 w0i w0j þ E26 w0i wj þ wi w0j þ E66 wi wj dz 0 Z lh i ¼ E23 w0i w0j þ E27 w0i wj þ E36 wi w0j þ E67 wi wj dz 0 Z lh ¼ E24 w0i w0j þ ðE28 E25 Þw0i wj þ E46 wi w0j 0 i þ ðE68 E56 Þwi wj dz Z lh i ¼ E33 w0i w0j þ E37 w0i wj þ wi w0j þ E77 wi wj dz 0 Z lh ¼ E34 w0i w0j þ ðE38 E35 Þw0i wj þ E47 wi w0j 0 i þ ðE78 E57 Þwi wj dz Z lh ¼ E44 w0i w0j þ ðE48 E45 Þ w0i wj þ wi w0j 0 i þ ðE88 2E58 þ E55 Þwi wj dz ð43Þ
In Eq. (42), the force vector is given by Z l fi3 ¼ qwi dz
l
E77 w0i w0j dz 0
ð44Þ
0
l
ðE57 þ E78 Þw0i w0j dz
This completes the finite element formulation of the present theory.
0
K 35 ij
Z lh i ¼ E27 w0i w0j þ E67 w0i wj dz
K 36 ij
Z lh i ¼ E37 w0i w0j þ E77 w0i wj dz
0
8. Numerical results and discussion
0
For the verification purpose, the results by present approach are compared with the previous results [6,8] Table 1 Deflections of a simply-supported I-section beam under uniformly distributed load Stacking sequence
ABAQUS
Ref. [6]
[0]16 [15/15]4s [30/30]4s [45/45]4s [60/60]4s [75/75]4s [0/90]4s
6.340 6.989 9.360 13.479 17.023 18.490 9.400
6.283 6.941 9.322 13.449 16.993 18.453 9.349
Ref. [8]
Present
s = 0
rs = 0
s = 0
rs = 0
6.103 6.610 8.281 11.340 15.119 17.643 9.153
6.233 6.899 9.290 13.421 16.962 18.411 9.299
6.129 6.637 8.307 11.363 15.151 17.683 9.189
6.259 6.923 9.314 13.446 16.992 18.449 9.381
220
J. Lee / Composite Structures 70 (2005) 212–222 10cm
9.0
0.5cm orthotropy (no shear)
8.0
FEM (no shear) orthotropy (shear)
7.0
FEM (shear)
6.0
0.5cm
20cm
5.0 v 4.0 3.0 2.0
Fig. 3. Thin-walled I-section composite beam.
1.0
and ABAQUS [10]. In ABAQUS analysis, S9R5 shell elements are used. A simply-supported I-section beam with 2.5 m length under uniformly distributed load of 1 kN/m with dimension of (50 · 50 · 2.08) mm is considered. The following engineering constants are used. E1 ¼ 53:78 GPa;
E2 ¼ E3 ¼ 17:93 GPa
G23 ¼ 3:45 GPa; G12 ¼ G13 ¼ 8:96 GPa m23 ¼ 0:34; m12 ¼ m13 ¼ 0:25
ð45Þ
The maximum vertical deflections based on two different assumptions (rs = 0 and s = 0) are compared with the previous results by Barbero [6], results by the author using classical beam model [8], and ABAQUS [10] in Table 1 for several stacking sequences. All the results are in good agreement. Generally, the results with
0.0 0
15
30
45 fiber angle
60
75
90
Fig. 5. Central deflection with respect to the fiber angle of a simplysupported anti-symmetric angle-ply I-section composite beam with l/b3 = 20 under uniformly-distributed load.
plane stress assumption (rs = 0) show excellent agreement with ABAQUS solution. For all the stacking sequences considered, the present approach exhibits better results than that of classical beam assumption. As a next example, a thin-walled I-section composite beam is considered in order to investigate the coupling and transverse shear deformation effects. The geometry of the section of the beam is shown in Fig. 3, and the following material properties are considered in this investigation: E1 =E2 ¼ 25 G12 =E2 ¼ 0:5 G13 =E2 ¼ 0:6
9.0 orthotropy (no shear)
8.0
m12 ¼ 0:25
FEM (no shear) orthotropy (shear)
7.0
ð46Þ
For convenience, the following nondimensional value of vertical displacements are used:
FEM (shear)
6.0
v ¼
5.0 v
vE2 b33 ql4
ð47Þ
4.0
Table 2 Ratio of coupling stiffnesses with respect to the bending stiffness when the top and bottom flanges, and the web are all antisymmetric angleply
3.0 2.0 1.0 0.0
0
15
30
45 fiber angle
60
75
90
Fig. 4. Central deflection with respect to the fiber angle of a simplysupported anti-symmetric angle-ply I-section composite beam with l/b3 = 50 under uniformly-distributed load.
Fiber angle
E15/E33
E27/E33
E38/E33
0 15 30 45 60 75 90
0.000 0.023 0.042 0.042 0.022 0.007 0.000
0.000 0.023 0.042 0.042 0.022 0.007 0.000
0.000 0.002 0.004 0.004 0.002 0.001 0.000
J. Lee / Composite Structures 70 (2005) 212–222
221
0.6
The beam is assumed to be under uniformly-distributed loading with simply-supported boundary condition. The top and bottom flanges and the web are considered as antisymmetric angle-ply laminates [h/h]. For all the analyses, the assumption rs = 0 is made. Variation of the deflection at the center of the beam with respect to the fiber angle change is shown in Figs. 4 and 5 for l/b3 = 50 and l/b3 = 20. In generating Figs. 3 and 4, the finite element solution with no shear is calculated based on previous paper [7]. The closed-form solution of the maximum deflection for the uncoupled equations (Eq. (39)) can be directly calculated for the beam with uniformly-distributed loading and simply-supported boundary conditions as [11]:
0.5
vmax ¼
Table 3 Ratio of coupling stiffnesses with respect to the bending stiffness when the top flange and web is unidirectional while the bottom flange is symmetric angle-ply Fiber angle
E13/E33
E24/E33
E68/E33
0 15 30 45 60 75 90
0.000 0.894 3.751 5.196 5.537 5.611 0.000
0.000 0.001 0.003 0.004 0.005 0.005 0.000
0.000 0.274 0.804 0.789 0.634 0.544 0.000
0.4
v 0.3
0.2 orthotropy (no shear) FEM (no shear)
0.1
orthotropy (shear) FEM (shear)
0.0
0
15
30
45 fiber angle
60
75
90
Fig. 6. Central deflection with respect to the fiber angle of a simplysupported I-section composite beam with symmetric angle-ply for the bottom flange and l/b3 = 50 under uniformly-distributed load.
0.6
0.5
0.4
v 0.3
0.2 orthotropy (no shear) FEM (no shear)
0.1
orthotropy (shear) FEM (shear)
0.0
0
15
30
45 fiber angle
60
75
90
Fig. 7. Central deflection with respect to the fiber angle of a simplysupported I-section composite beam with symmetric angle-ply for the bottom flange and l/b3 = 20 under uniformly-distributed load.
5ql4 ql2 þ 384ðEI x Þcom 8ðGAx Þcom
ð48Þ
In Eq. (48), the first term denotes the deflection by classical beam theory, and the second term is the deflection by shear deformation. For this stacking sequence, the coupling stiffnesses E15, E27 and E38 do not vanish while all the other coupling stiffnesses become zero. That is, the orthotropy solution given in Eq. (48) might not be accurate. However, the coupling stiffnesses are very small compared to bending stiffness E33 (Table 2). Consequently, the coupling effects coming from the material anisotropy become negligible for two different spanto-height ratios. It also shows that the transverse shear effects are also negligbly small even for the beam with the lower span-to-height ratio (l/b3 = 20) indicating that the simple orthotropy solution of the classical beam theory is sufficiently accurate for these cases. To investigate the coupling and transverse shear effects further, the same configuration with the previous example except the laminate stacking sequence is considered. The stacking sequence of the top flange and the web of the beam is unidirectional while the bottom flange is assumed to be [h/h/h/h]. For this stacking sequence, the coupling stiffnesses E13, E24 and E68 do not vanish while all the other coupling stiffnesses become zero. Especially, E13 becomes no more negligibly small as given in Table 3. For l/b3 = 50, as fiber angle increases, the orthotropic and finite element solutions show discrepancy indicating the coupling effects become significant (Fig. 6). The transverse shear little affects the behavior of the beam. For lower span-to-height ratio (Fig. 7), however, the solutions excluding shear effects remarkably underestimate the deflection for all the range of fiber angle. Furthermore, the orthotropic solutions disagree with the finite element solutions as orthotropy of the beam gets higher.
222
J. Lee / Composite Structures 70 (2005) 212–222
9. Concluding remarks
References
An analytical model was developed to study the flexural behavier of a laminated composite beam with an I-section. The model is capable of predicting accurate deflection for various configuration including boundary conditions, laminate orientation and span-to-height ratio. To formulate the problem, a one-dimensional displacement-based finite element method is employed. The assumption that normal stress in contour direction vanishes (rs = 0) seems more appropriate than free strain assumption in contour direction (s = 0). The shear effects become significant for lower span-to-height ratio and higher degrees of orthotropy of the beam. The orthotropy solution is accurate for lower degrees of material anisotropy, but, becomes inappropriate as the anisotropy of the beam gets higher, and fully coupled equations should be considered for accurate analysis of thin-walled composite beams. The presented analytical model is found to be appropriate and efficient in analyzing flexural problem of thin-walled laminated composite beams.
[1] Vlasov VZ. Thin-walled elastic beams. second ed. Jerusalem, Israel: Israel Program for Scientific Translation; 1961. [2] Gjelsvik A. The theory of thin-walled bars. New York: John Wiley and Sons Inc.; 1981. [3] Bauld NR, Tzeng LS. A Vlasov theory for fiber-reinforced beams with thin-walled open cross section. Int J Solids Struct 1984;20(3):277–97. [4] Davalos JF, Salim HA, Qiao P, Lopez-Anido R. Analysis and design of pultruded FRP shapes under bending. Compos: Part B 1996;27B:295–305. [5] Ascione L, Feo L, Mancusi G. On the statical behavior of fiberreinforced polymer thin-walled beams. Compos: Part B 2000;31B: 643–654. [6] Barbero EJ. Introduction to composite materials design. Taylor & Francis; 1999. [7] Qin Z, Librescu L. On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations. Compos Struct 2002;56(2):345–58. [8] Lee J, Lee S. Flexural–torsional behavior of thin-walled composite beams. Thin-walled Struct 2004;42(9):1293–305. [9] Jones RM. Mechanics of composite materials. Hemishere Publishing Corporation; 1975. [10] ABAQUS/Standard UserÕs Manual, Version 6.4, Hibbit, Kalsson & Sorensen Inc.; 2003. [11] Reddy JN. Mechanics of laminated composite plates: theory and analysis. CRC Press; 1997.
Acknowledgment The support of the research reported here by Korea Research Foundation through Grant KRF-2003-041D00595 is gratefully acknowledged.