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Static analysis of thin-walled laminated composite closed-section beams with variable stiffness
Composite Structures 182 (2017) 67–78
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Static analysis of thin-walled laminated composite closed-section beams with variable stiffness M. Gökhan Günay, Taner Timarci ⇑ Trakya University, Mechanical Engineering Department, Edirne 22180, Turkey
a r t i c l e
i n f o
Article history: Received 4 July 2017 Revised 15 August 2017 Accepted 29 August 2017 Available online 1 September 2017 Keywords: Composite materials Thin-walled beams Variable stiffness Finite element analysis
a b s t r a c t Static behavior of thin-walled laminated composite closed cross-section beams having variable stiffness is investigated in this study. The analytical model used accounts for flexural–torsional coupling and warping effects as well as the variable stiffness along the contour of the cross-section of the beam. The variable stiffness is acquired by constructing the laminates with curvilinear fibres having certain specific paths. The orientation of fibres varies by depending on the fibre path along the contour of the crosssection in each layer. Equilibrium equations are derived by use of minimum potential energy principle. Although the formulation given can be applied to any shape of the closed cross-section with straight or curved edges, preliminary numerical results are presented only for box-beams. A displacement based finite element method is developed to solve the analytical model and to predict displacements and rotations under the effect of different types of loading conditions. Numerical results are obtained for different fibre paths and lay-up configurations and compared with the available solutions in the literature also with the results of a finite element analysis software using shell element. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Thin walled composite laminated beams are widely used as structural elements in many engineering applications. The structures made of composite materials have many advantages such as having the high specific modulus and strength compared to the structures made of conventional materials. One other important advantage of composite layered structures reinforced with continuous fibres is that they can be tailored according to the requirements of their application area by changing, for example, thicknesses and the stacking sequences of layers also the orientation of fibres in layers. The theoretical background of thin-walled members are mostly based on the works of Vlasov [1] and Gjelsvik [2] where the beams are made of isotropic materials. There are many researches realised for the understanding the mechanical behavior of thin-walled beams made of composite materials. The coupling effects for symmetric and anti-symmetric box-beams under flexural, torsional, and extensional loads are presented in the work of Chandra et al. [3]. Song and Librescu [4] formulated and investigated the free vibration behavior of laminated composite thick and thin walled, single-cell beams of arbitrary crosssections. A beam theory including expressions for the stiffness ⇑ Corresponding author. E-mail address: [email protected] (T. Timarci). http://dx.doi.org/10.1016/j.compstruct.2017.08.092 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved.
matrix for thin-walled open and closed-section composite beams with arbitrary lay-ups was developed by Kollar and Pluzsik [5]. Piovan and Cortinez [6] presented a theoretical model for the dynamic, static and buckling analyses of closed cross-section composite thin-walled beam by taking into account the shear flexibility due to bending and warping. Vo and Lee [7] developed a general analytical model for flexural-torsional behavior of thin-walled composite box-beams. The same authors extended their approach for the geometric nonlinearity [8]. Sheikh and Thomson [9] adopted a closed-form approach to derive the cross-sectional stiffness matrix for open and closed-section composite beams taking into account the coupling effects and used a three-node beam element in the finite element analysis. a one-dimensional beam model considering the effects of axial warping and material anisotropies for single-cell composite beams was developed by Zhang and Wang [10]. Carrera et al. [11] realised the static analysis of laminated beams with thin-walled cross-sections in the frame of Carrera Unified Formulation (CUF) based on Taylor (TE) and Lagrange Expansions (LE) of cross-sectional functions, leading to accurate and computationally efficient finite element methods. Composite structures are generally produced by using straight and evenly distributed fibres. An additional useful property can be added to the composite laminated structures by using curvilinear fibres in layers instead of using conventional straight fibres. This provides another flexibility in design of such structures with
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
variable stiffness laminates to satisfy certain mechanical requirements. The mechanical properties of structural elements can also be varied by changing the distribution or the shape of fibres. Martin and Leissa [12] played with the distance between fibres to achieve a variable stiffness composite plate and applied Ritz method to several plane elasticity problems. The curved fibres were used by Gürdal and Olmedo [13] to vary stiffnesses of rectangular composite plates. They analyzed the elastic response of the variable stiffness panel by use of closed-form and numerical solutions. It was also shown by Gürdal et al. [14] that it is possible to change either the buckling load or the in-plane stiffness while keeping the other one constant for variable stiffness panels. Variable stiffness concept is investigated commonly on plates and cylinders but rarely on thin-walled beams. In the paper of Akhavan at al. [15] large deflections and stresses of variable stiffness composite laminated plates with curvilinear fibres were calculated and it was shown that the fibre variation may be used to reduce deflections and stresses in some static loadings. Zamani et al. [16] used curvilinear fibres to optimize thin-walled beams with bi-convex cross section. In the present study the static behavior of thin-walled composite beams with variable stiffness and thickness is investigated. The formulation developed for thin-walled composite beams with closed cross-section is based on the theory given in Librescu and Song [17]. On the basis of this formulation the present model accounts for the variable stiffness as well as the variation of thickness in layers. A mapping procedure to facilitate the usage of different fibre angle variations in layers is proposed, furthermore a new description and lay-up code is also used to define variable stiffness layers on thin-walled composite beams in a similar manner of straight fibres case. Although the formulation presented can be applied to any shape of the closed cross-section with straight or curved edges, preliminary numerical results are presented only for the composite box-beams. A displacement based finite element method which is parallel to the presented in Vo and Lee [7] is used to solve the analytical model and to predict displacements and rotations under the effect of different types of loading conditions. Numerical results obtained for different fibre paths and lay-up configurations are presented and compared with the results of a commercial finite element analysis software in which the shell element is utilized. 2. Kinematics The two coordinate systems used are shown in Fig. 1. The first coordinate system is the global cartesian (x, y, z) coordinate system. The x- and y-axes lie in the plane of the cross-section and the z-axis is along the longitudinal axis of the beam. The second one is the local coordinate system (n, s, z) where n-axis is normal
to the middle surface as s-axis is tangent to the middle surface and it is directed along the contour line of the cross-section. These two coordinate systems are related through an angle of orientation a. Point P called as the pole point is used as a reference to define displacements and rotations of cross section. U and V are displacements of the pole point in the global coordinate system along x and y axes. / is the rotation of cross section around the pole point. 2.1. Assumptions The following assumptions are made in developing the kinematical relations. (i) The cross section of beam is assumed to maintain its shape in its own plane during deformation. There is no cross sectional distortion although it is allowed to warp out of its own plane. (ii) The normals to the undeformed middle surface remain normal to the deformed middle surface. (iii) The ratio of the wall thickness to the radius of curvature at any point of the beam wall is negligibly small compared to unity. (iv) The hoop stress rs in the contour direction s is small compared to the axial stress rz . (v) Transverse shear strains cxz and cyz are neglected. 2.2. Displacement field Displacements of an arbitrary point m(x,y) on the middle surface can be expressed by the displacements of point P(xp, yp) in the local coordinate system as given below [16]:
n ðs; zÞ ¼ UðzÞ u
dy dx VðzÞ /ðzÞ r s ds ds
ð1aÞ
s ðs; zÞ ¼ UðzÞ u
dx dy þ VðzÞ þ /ðzÞ r n ds ds
ð1bÞ
n; zÞ ¼ W 0 ðzÞ U 0 ðzÞ xðsÞ V 0 ðzÞ yðsÞ /0 ðzÞ wðsÞ wðs;
ð1cÞ
where W 0 is average axial displacement of cross section, wðsÞ is primary warping function. r n ðsÞ is the normal component and r s ðsÞ is the tangential component of the distance between points P and m as described below:
rn ðsÞ ¼ ðxðsÞ xp Þ
dy dx ðyðsÞ yp Þ ds ds
ð2aÞ
rs ðsÞ ¼ ðxðsÞ xp Þ
dx dy þ ðyðsÞ yp Þ ds ds
ð2bÞ
Similarly displacements of an arbitrary point M (X,Y) laying out of the middle surface can be defined in the local coordinate system as:
U n ðs; n; zÞ ¼ UðzÞ
dy dx VðzÞ /ðzÞ Rs ðs; nÞ ds ds
ð3aÞ
U s ðs; n; zÞ ¼ UðzÞ
dx dy þ VðzÞ þ /ðzÞ Rn ðs; nÞ ds ds
ð3bÞ
dy dx V 0 ðzÞ yðsÞ n Wðs; n; zÞ ¼ W 0 U 0 ðzÞ xðsÞ þ n ds ds þ n wðsÞ /0 ðzÞ ½wðsÞ
Fig. 1. Displacements of an arbitrary point (m) on the middle surface.
ð3cÞ
where wðsÞ is the secondary warping function, Rn ðs; nÞ is the normal component and Rs ðs; nÞ is the tangential component of the distance
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where primes ()’ denote the derivative with respect to z. ezz and csz strains can be separated into to middle surface strains ()(0) and middle surface curvatures ()(1) as given below: ð1Þ ezz ¼ eð0Þ zz þ n ezz
ð9aÞ
ezzð0Þ ¼ w00 U00 x V00 y /00 w
ð9bÞ
00 eð1Þ zz ¼ U
dy dx þ V00 /00 wðsÞ ds ds
ð1Þ csz ¼ cð0Þ sz þ n csz
cszð0Þ ¼
Fig. 2. An arbitrary point (M) out of the middle surface.
ð9cÞ ð10aÞ
2X ; hðsÞ Gsz ðsÞ L
cszð1Þ ¼
2b hðsÞ Gsz ðsÞ L
ð10bÞ
between points P and M as shown in Fig. 2. They can be expressed by the terms rn ðsÞ and r s ðsÞ as:
2.4. Governing equations
Rn ðs; nÞ ¼ r n ðsÞ þ n
ð4aÞ
Rs ðs; nÞ ¼ r s
ð4bÞ
Equilibrium equations of the thin-walled beam can be obtained by equating the variation of the total potential energy of the system to zero. The total potential energy of the system, Ptotal , is the sum of strain energy, Pstrain , and work done by external forces and moments, W ext as, given below:
Definitions of the primary and secondary warping functions are given below:
Z s wðsÞ ¼ rn 0
Z s wðsÞ ¼
2X ds hðsÞ Gsz ðsÞ L
ð5aÞ
2X ¼
I
I
r
ðsÞ ds;
b¼
I
d
L¼
s;
ds hðsÞGsz ðsÞ
Z 1 ðrðkÞ ezz þ rðkÞ sz csz Þdv 2 v zz
ð5bÞ
where hðsÞ is the thickness of the cross-section, Gsz ðsÞ is the effective shear stiffness of the laminate. 2X, b being the double area enclosed by the center line of the beam cross-section and its perimeter, respectively and L are defined as follows:
ð11Þ
The strain energy is:
Pstrain ¼
b ds rs hðsÞ Gsz ðsÞ L
Ptotal ¼ Pstrain þ W ext
By substituting Eqs. (9) and (10) into Eq. (12) the strain energy can be obtained as:
Z 1 dy dx V 00 ðy n Þ rzzðkÞ ðsÞ W 0 0 U 00 x þ n 2 v ds ds 1 2 X 2b /00 wðs; nÞ þ rðkÞ /0 dv þn sz ðsÞ hðsÞ Gsz ðsÞ L L
Pstrain ¼
ð6Þ
ð13Þ
n
For a thin-walled laminated composite beam the effective shear stiffness Gsz ðsÞ given in [18] can be calculated as expressed below:
"
2
1 a16 ðsÞ Gsz ðsÞ ¼ a66 ðsÞ hðsÞ a11 ðsÞ
#
ð7Þ
where aij ðsÞ are modified beam stiffnesses which are defined explicitly in Appendix A.
This equation can be rearranged as:
ezz
dy dx ¼ W 0 0 U 00 x þ n V 00 y n /00 wðs; nÞ ds ds
0
l
ðNz W 00 M x V00 M y U00 þ M z /0 M x /00 Þdz ð14Þ
where the beam forces and the moments are defined as:
Z
Nz ¼
Nzz ðsÞ ds
ð15aÞ
Z Mx ¼
Z
ð8eÞ
x Nzz ðsÞ Lzz ðsÞ
dy ds ds
ð15cÞ
ðsÞ Lzz ðsÞ xðsÞds Nzz ðsÞ x
ð15dÞ
c
Z Mz ¼
ð8dÞ
ð15bÞ
c
ð8bÞ
2X 2b csz ¼ þn hðsÞ Gsz ðsÞ L hðsÞ Gsz ðsÞ L
dx ds ds
Z My ¼ Mx ¼
ð8cÞ
y Nzz ðsÞ þ Lzz ðsÞ c
ð8aÞ
@W @U s þ csz ¼ ds @z
@W @U n þ cnz ¼ ¼0 @n @z
Z
c
By the assumption (i) in-plane strains will be zero (exx ¼ 0, eyy ¼ 0, exy ¼ 0). Axial strain ezz and shear strains csz and cnz can be calculated with small displacements assumption as:
ezz
1 2
Pstrain ¼
2.3. Strain field
dW ¼ dz
ð12Þ
Nsz ðsÞ c
2X 2b Lsz ðsÞ ds hðsÞ Gsz ðsÞ L hðsÞ Gsz ðsÞ L
ð15eÞ
Here N ab ðsÞ and Lab ðsÞ (as a,b = s,z) are shell stress resultants are couples along the thickness of the laminate respectively as given below and as shown in Fig. 3.
Nab ðsÞ ¼
N Z X k
n;k
n;k1
rðkÞ ab dn
ð16aÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Fig. 3. (a) Shell stress resultants and (b) couples.
Lab ðsÞ ¼
N Z X k
n;k
n;k1
ðkÞ
n rab dn
ð16bÞ
The work done by external forces and moments is given as:
Z W ext ¼ 0
l
ðqz W 0 þ qx U þ qy V þ mz /Þdz
ð17Þ
where qx and qy are transverse loads, qz is axial load and mz is axial torque. By equating the variation of the total potential energy to zero, the following weak formulation can be obtained:
dPtotal ¼ dPstrain þ dW ext ¼ 0 Z 0¼ 0
l
nesses. Calculations of these variable stiffness elements are explained in next section. Based on the assumption (iv) transverse stress resultant N ss ðsÞ and transverse stress couple Lss ðsÞ will be very small compared to other stress resultants and couples. By taking these hoop terms as zero as below:
essð0Þ ¼
þ qz dW 0 þ qx dU þ qy dV þ mz d/Þdz
þ ðA26 ðsÞ D22 ðsÞ B22 ðsÞ B26 ðsÞÞcð0Þ zz þ ðB26 ðsÞ D22 ðsÞ ð1Þ B22 ðsÞ D26 ðsÞÞczz
eð1Þ ss ¼ ð19Þ
ð20aÞ
M 00y þ qx ¼ 0
ð20bÞ
M 00x þ qy ¼ 0
ð20cÞ
M 00w þ M0z þ mz ¼ 0
ð20dÞ
2.5. Constitutive equations The constitutive relations between shell stress resultants and couples and strains of a composite shell consisting of the layers with variable stiffness are defined as follows:
3 2 3 A11 ðsÞ A12 ðsÞ A16 ðsÞ B11 ðsÞ B12 ðsÞ B16 ðsÞ N zz ðsÞ 6 N ðsÞ 7 6 A ðsÞ A ðsÞ A ðsÞ B ðsÞ B ðsÞ B ðsÞ 7 22 26 12 22 26 7 6 ss 7 6 12 7 6 7 6 6 N sz ðsÞ 7 6 A16 ðsÞ A26 ðsÞ A66 ðsÞ B16 ðsÞ B26 ðsÞ B66 ðsÞ 7 7 6 7 6 6 L ðsÞ 7 ¼ 6 B ðsÞ B ðsÞ B ðsÞ D ðsÞ D ðsÞ D ðsÞ 7 12 16 11 12 16 7 6 11 7 6 zz 7 6 7 6 4 Lss ðsÞ 5 4 B12 ðsÞ B22 ðsÞ B26 ðsÞ D12 ðsÞ D22 ðsÞ D26 ðsÞ 5 2
Lsz ðsÞ
B16 ðsÞ B26 ðsÞ 2 ð0Þ 3
B66 ðsÞ
D16 ðsÞ D26 ðsÞ D66 ðsÞ
ezz
1 A22 ðsÞ D22ðsÞ B222 ðsÞ
ð22aÞ ½ðA12 ðsÞ B22 ðsÞ B12 ðsÞ A22 ðsÞÞeð0Þ zz
þ ðB12 ðsÞ B22 ðsÞ A22 ðsÞ D12 ðsÞÞeð1Þ zz þ ðA26 ðsÞ B22 ðsÞ A22 ðsÞ B26 ðsÞÞcð0Þ zz þ ðB26 ðsÞ B22 ðsÞ
Equilibrium equations of beam can be obtained by integrating derivatives of varied quantities by parts in Eq. (19) and collecting the coefficients of dw0 , dU, dV and d/ as:
N0z þ qz ¼ 0
½ðA12 ðsÞ D22 ðsÞ B12 ðsÞ B22 ðsÞÞeð0Þ zz
þ ðB12 ðsÞ D22 ðsÞ B22 ðsÞ D12 ðsÞÞeð1Þ zz
ð18Þ
ðNz dW 00 Mx dV 00 M y dU 00 þ M z d/0 Mx d/00
1 A22 ðsÞ D22ðsÞ B222 ðsÞ
ð1Þ eð0Þ ss and ess strains can be expressed
A22 ðsÞ D26 ðsÞÞcð1Þ zz
ð22bÞ
By substituting Eqs. (22) into Eq. (21) the modified constitutive relations can be expressed as given below:
3 2 a ðsÞ a ðsÞ b ðsÞ 11 16 11 Nzz ðsÞ b 6 N ðsÞ 7 6 ðsÞ a ðsÞ b a 16 66 6 sz 7 6 16 ðsÞ 7¼6 6 6 4 Lzz ðsÞ 5 4 b ðsÞ bb ðsÞ d ðsÞ 11 11 16 a Lsz ðsÞ b ðsÞ b66 ðsÞ d16 ðsÞ 2
16
a
b16 ðsÞ
3 2
3
eð0Þ zz 7 6 ð0Þ 7 6 csz 7 b66 ðsÞ 7 76 7 7 6 ð1Þ 7 d16 ðsÞ 5 4 ezz 5 cð1Þ d66 ðsÞ sz
ð23Þ
The detailed expressions for the stiffness coefficients aij ðsÞ bij ðsÞ and dij ðsÞ are given in Appendix A. The constitutive relations between the beam forces and moments and the derivatives of the pole point displacement terms can be obtained by substituting modified constitutive relations into Eqs. (15) as:
3 2 e11 Nz 7 6 6 6 My 7 6 7 6 6 6 Mx 7 ¼ 6 7 6 6 7 6 6 4 Mx 5 4 Mz 2
e12
e13
e14
e22
e23
e24
e33 sym
e34 e44
3 W 00 7 6 7 e25 7 6 U 00 7 7 6 7 00 7 6 e35 7 7 6 V 7 7 6 7 e45 5 4 /00 5 0 e55 / e15
3 2
ð24Þ
where ½eij is beam stiffness matrix of thin-walled composite laminated beam with variable stiffness. Elements of ½eij can be expressed as:
Z
6 ð0Þ 7 6 ess 7 7 6 6 cð0Þ 7 6 sz 7 6 ð1Þ 7 6 ezz 7 7 6 6 ð1Þ 7 4 ess 5
e11 ¼
a11 ðsÞ ds
ð25aÞ
s
e12
cszð1Þ ð21Þ Here, the elements of A, B, D matrices are the functions of s along the beam cross-sectional contour which differ from the conventional definitions of extensional, coupling and bending stiff-
e13
Z dy ds ¼ a11 ðsÞ xðsÞ þ b11 ðsÞ ds s Z dx ds ¼ a11 ðsÞ yðsÞ b11 ðsÞ ds s Z
e14 ¼ s
1 2X 2b a ds a16 ðsÞ þ b16 ðsÞ hðsÞ Gsz ðsÞ L L
ð25bÞ ð25cÞ
ð25dÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Z e15 ¼
ðsÞ þ b11 ðsÞ xðsÞÞ ds ða11 ðsÞ x
ð25eÞ
iv iv iv III e13 W III 0 e23 U e33 V e34 / þ e35 / þ qy ¼ 0
ð26cÞ
s
Z e22 ¼ s
2 ! dy dy þ d11 ðsÞ a11 ðsÞ xðsÞ þ 2 b11 ðsÞ xðsÞ ds ds ds ð25fÞ
e23
Z dy dx ¼ ða11 ðsÞ xðsÞ yðsÞ þ b11 ðsÞ yðsÞ xðsÞ ds ds s dx dy ds d11 ðsÞ ds ds 1 2X 2b a a16 ðsÞ xðsÞ þ b16 ðsÞ xðsÞ L L s hðsÞ Gsz ðsÞ 2X dy 2b dy b þ d16 ðsÞ ds þ b16 ðsÞ L ds L ds
ð25gÞ
Z
e24 ¼
e25 ¼
ð25hÞ
Z dy ðsÞ þ b11 ðsÞ ðx ðsÞ þ xðsÞ xðsÞÞ a11 ðsÞ xðsÞ x ds s dy ds ð25iÞ þ d11 ðsÞ xðsÞ ds Z
e33 ¼ s
2 ! dx dx 2 ds a11 ðsÞ yðsÞ 2 b11 ðsÞ yðsÞ þ d11 ðsÞ ds ds ð25jÞ
1 2X 2b a a16 ðsÞ yðsÞ þ b16 ðsÞ yðsÞ L L s hðsÞ Gsz ðsÞ 2X dx 2b dx b ds b16 ðsÞ d16 ðsÞ L ds L ds Z
e34 ¼
e35 ¼
ð25kÞ
Z dx ðsÞ b11 ðsÞ ðx ðsÞ yðsÞ xðsÞÞ a11 ðsÞ yðsÞ x ds s dx ds ð25lÞ d11 ðsÞ xðsÞ ds 2X 2X 2X 2b a66 ðsÞ þ 2 b66 ðsÞ 2 2 L L L L s hðsÞ Gsz ðsÞ 2b 2b ds ð25mÞ þ d66 ðsÞ L L Z
e44 ¼
1
1 2X 2b a ðsÞ ðsÞ a16 ðsÞ x þ b16 ðsÞ x L L s hðsÞ Gsz ðsÞ 2X 2b b ds þ d16 ðsÞ xðsÞ þb16 ðsÞ xðsÞ L L Z
e45 ¼
e55 ¼
ð25nÞ
Z ðsÞ2 þ b11 ðsÞ x ðsÞ xðsÞ þ d11 ðsÞ xðsÞ2 ds a11 ðsÞ x s
ð25oÞ Diagonal elements of ½eij are primary stiffnesses related to the extension along z-axis, to the bending about y- and x-axes and to the bi-moment and torsion about z-axis. Remaining elements of ½eij are the stiffnesses of the coupled behaviors. By substituting Eq. (24) into Eqs. (20) the explicit form of the governing equations are obtained in terms of these beam stiffnesses and the derivatives of pole displacements as:
e11
W II0
e12
W III 0
II iv III iv III e14 W III 0 þ e15 W 0 e24 U e25 U e34 V e35 V
e44 /iv þ 2e45 /III þ e55 /II þ mz ¼ 0
2
e12 U
IIIe13
iv
III
III
II
V e14 / þ e15 / þ qz ¼ 0 iv
iv
III
e22 U e23 V e24 / þ e25 / þ qx ¼ 0
ð26aÞ ð26bÞ
ð26dÞ
3. Description of the variable stiffness Stiffnesses of a continuous fibre reinforced composite layer depends on the orientation angle of the fibres. A desired orientation angle variation can be achieved by placing curvilinear fibres to follow a specific path. Thus stiffness of a layer produced with curvilinear fibres will be spatially varying depended on the position and the direction. A simple procedure is applied in order to facilitate in handling beams composed of variable stiffness layers with different configurations. In this procedure the variable stiffness layers are defined on a new f g plane separately corresponding to the local s-z plane and a name is given to each layer. Fibre orientation angle variation of a variable stiffness layer is expressed in terms of these coordinates same as the one presented in the work of Gürdal and Olmedo [13] formulation by using a linear interpolation as:
hðfÞ ¼
ðh1 h0 Þ f þ h1 ; 1 6 f 6 0 ðh2 h1 Þ f þ h1 ; 0 < f 6 1
ð27Þ
where h1 expresses the orientation angle at the origin, h0 and h2 express the orientation angles at f ¼ 1 and f ¼ 1 respectively. This orientation angle variation is satisfied by the following the fibre path definition: 8 ln½cosððh0 h1 Þ f þ 90 h1 Þ þ ln½cosð90 h1 Þ > > ; 1 6 f 6 0 > h1 Þ < p ðh0180 gðfÞ ¼ > ln½cosððh1 h2 Þ f þ 90 h1 Þ þ ln½cosð90 h1 Þ > > ; 0
ð28Þ As an example, a variable stiffness layer named as ‘‘a” is defined in Table 1. Fibre orientation angle variation, the corresponding fibre path and the change of the thickness is also shown in Fig. 4. A layer with variable stiffness defined in such a way can be mapped on a layer of any desired segment of the thin-walled composite beam as shown in Fig. 5. In Table 2 it is seen that f g alignments can be done in four different ways by flipping f g plane. The mapping of the variable stiffness layer- a on the kth layer of segment-i is given for each alignment where si1 and si denote the borders of the ith segment. Symbolic representation of the alignments for layer-a are expressed by ðfÞ ðgÞ a, where (+) sign means the default direction and () means the flipped direction of the corresponding axes. It is assumed that the variable stiffness layers are produced by shifting the fibre path along g axis and placing the new fibres next to the previous ones such that any gap or overlap among layers is not allowed to occur. As a result, the stiffness of the layer will be constant along g axis while will be varying along f axis. An experimental production method called as Continuous Tow Shearing [19,20] developed recently offers the capability to fulfilling these requirements while the other methods such as Automated and Tailored Fibre Placement cause the gaps or overlaps in the structure. On the
Table 1 Definition of variable stiffness layer-a. Layer
h0
h1
h2
h0
href
a
700
200
700
1 mm
200
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Fig. 4. (a) Orientation angle variation, (b) corresponding fibre path and (c) thickness change of variable stiffness layer-a.
Fig. 5. Mapping of variable stiffness layer on the first segment of the thin-walled composite beam.
other hand in Continuous Tow Shearing method the thickness of the produced layer varies by depending on the change of the orientation angle. This relation can be expressed as follows:
here nk ðsÞ terms will depend on the thickness change of layers in the lay-up.
hðfÞ ¼ h0 =cosðhðfÞ href Þ
4. Finite element method
ð29Þ
Here href is the reference angle of the region where the thickness of the layer is equal to the original thickness (ho ). Upon mapping the elements of A(s), B(s), D(s) matrices are calculated by using the well known formulations [21] depending on hðsÞk as:
Aij ðsÞ ¼
N X ij ðhðsÞ Þ:ðnk ðsÞ nk1 ðsÞÞ Q k
ð30aÞ
k¼1
The governing equations are solved with displacement based finite element method by following similar procedure presented in [7]. 2-node beam elements are used to discretize thin-walled beam. Each node has seven degrees of freedom; three for translation, three for rotation and one for the rate of twist. Generalized displacements are expressed as:
Wo ¼
N X wi SL;i
ð31aÞ
i¼1
Bij ðsÞ ¼
Dij ðsÞ ¼
N 1X ij ðhðsÞ Þ:ðn2 ðsÞ n2 ðsÞÞ Q k k1 k 2 k¼1
ð30bÞ
N 1X ij ðhðsÞ Þ:ðn3 ðsÞ n3 ðsÞÞ Q k k1 k 3 k¼1
ð30cÞ
U¼
N X ui SH;i
ð31bÞ
i¼1
V¼
N X
v i SH;i
i¼1
ð31cÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78 Table 2 Alignment, symbolic representation and formulation of mapping variable stiffness layer-a to the kth layer of ith segment where 1 f 1 si1 s si .
U¼
N X /i SH;i
ð31dÞ
i¼1
½K fDg ¼ fFg
ð32Þ
The global stiffness matrix ½K and force vector fFg can be constructed by summing the associated terms of the beam elements sharing the same node and by using the element stiffness matrix ½k and element force vector ff g which are given as:
6 6 ½k ¼ 6 4
hðsÞk ¼ hðfÞa
þ a
hðsÞk ¼ hðfÞa
þa
hðsÞk ¼ hðfÞa
a
hðsÞk ¼ hðfÞa
5. Numerical examples 5.1. Straight fibres case
where SL;i is Lagrange linear shape function and SH;i is Hermite-cubic shape function associated with ith node. By substituting these expressions into Eq. (19) the finite element formulation is obtained in the matrix form as given below where ½K is global stiffness matrix, fFg is nodal force vector and fDg is nodal displacement vector:
2
þ þa
k11
k12
k13
k22
k23
k14
k24 7 7 7 k34 5
k33 sym:
ff g ¼ f f 1
3
ð33Þ
k44 f2
f3
T
f4 g
ð34Þ
Elements of ½k and ff g are given in explicitly in Appendix A. After applying the boundary conditions and the external forces, unknown nodal displacements are obtained by solving Eq. (32).
Validation of the present formulation is firstly realised by comparing the results obtained with the ones of previous works in which the conventional straight fibres in the layers of composite box-beam are used. The two cases of composite box-beams having material properties given in Table 3 and the geometry and the loading conditions shown in Figs. 6 and 7 same with Refs. [3,11,23,24] and [8,9,22] are considered in the numerical results obtained. Initially, for the case of Fig. 6 with Material-1, variations of the angle of twist along the axis of the beam subjected to 0.113 N m tip torque, with 6-layers for two circumferentially uniform stiffness (CUS2 and CUS3) lay-ups are obtained and compared with the experimental results of Chandra et al. [3] and the ones obtained by Kim and White [23] Qin and Librescu [24] and by both LE and TE models of Carrera et al. [11], as shown in Fig. 8. The beam CUS2 has a [00/300]3 stacking sequence in the top flange and the left web and [00/300]3 sequence in the bottom flange and the right web. For the lay-up CUS3, the only difference is that 450 fibre orientation angle in layers replaces 300 angle of CUS2. It can be said that the results of the present approach are in good agreement with the ones of the previous works as they are in between of the solutions of two models of CUF presented by Carrera et al.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
[8,9,22]. As seen from the table the results obtained are very close the ones of the previous works for the plane strain case.
Table 3 Composite material properties. Material-1 [3,11,23,24] Material-2 [8,9,22]
E11 = 142 GPa
E22 = 9,79 GPa
G12 = 6 GPa
v12 = 0,42
E11 = 148 GPa
E22 = 9,65 GPa
G12 = 4,55 GPa
v12 = 0,34
5.2. Curvilinear fibres case
[11] for CUS3 and closer to the ones of Refs. [23,24] for CUS2. For the case of the composite box-beam shown in Fig. 7 with Material-2, the beam is clamped at each end and loaded with 6,5 kN/m eccentric distributed load. Lay-up of each wall of the beam is [[45/45]2/012/[45/45]2] with total layer thickness of 2 mm. The results for maximum transverse displacement and axial rotation are given in Table 4 and compared with the ones of Refs.
As the main objective of the present work, the thin-walled boxbeam shown in Fig. 8 with the same geometry and loading and boundary conditions is considered for the variable stiffness case, again with Material-2 properties. The new lay-up of the beam is þ represented as ½ðþ þ b=þ bÞ2 =012 =ðþ b=þ bÞ2 and layer-b is defined in Table 5. Angles of h0 and h1 are changed from 00 to 900 by 150 and the values of maximum transverse displacement and axial rotation are obtained for each combination and presented in Table 6 where hh0 ¼ h1 ¼ 450 i case corresponds to the previous example of thin-walled beam with straight fibres. The thickness
Fig. 6. Clamped-free box-beam with a torque subjected at its tip.
Fig. 7. Eccentrically uniform distributed loaded box-beam.
Fig. 8. Angle of twist distribution along the cantilever box-beam.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78 Table 4 Maximum transverse displacement and maximum axial rotation of eccentrically loaded composite box-beam. Reference
V (mm)
U 103 (rad)
Sheikh & Thomsen (rs = 0) [9] Sheikh & Thomsen (es = 0) [9] Vo & Lee (rs = 0) [8] Vo & Lee (es = 0) [8] Kollar & Springer [22] Present
0,494 0,438 0,494 0,438 0,488 0,488
6,70 2,76 6,43 2,68 2,76 2,68
Table 5 Definition of variable stiffness layer-b. Layer
h0
h1
h2
h0
href
b
0:15:90
0:15:90
h2 ¼ h0
0,1 mm
href ¼ h1
of the layer goes to infinity as the difference between h0 and h1 reaches to 900. Therefore this kind of combinations are not taken into consideration. It is seen that in cases of hh0 ¼ h1 ¼ 00 i and hh0 ¼ 00 ; h1 ¼ 750 i the transverse displacements are smallest
when hh0 ¼ h1 ¼ 00 i and hh0 ¼ 00 ; h1 ¼ 750 i combinations lead to a higher and a smaller value for the axial rotation, respectively. It can be said that it is possible to have a better structural performance as required by designing laminated structures with variable stiffness providing more alternatives in their design compared to the ones with conventional straight fibres. For the same beam, the maximum transverse displacement and maximum axial rotation results of the variable stiffness beam by varying h0 and h1 values are obtained and given in Fig. 9 as contour plots showing the curves of the equivalent values for the displacements and rotations scaled by the values for hh0 ¼ h1 ¼ 450 i case of straight fibres. In these plots it is seen that same transverse displacement or axial rotation can be achieved with various values of h0 and h1 which leads to have a flexibility for the optimization process in designing the structure. As another application of the present formulation, the composite box-beams with variable stiffness, clamped from one end with different types of loading as axial Fz, transverse Fy and Fz as well as the torsional moment Mz applied separately and as their combination at its free end are considered as shown in Fig. 10. In this example the composite box-beams made of variable stiffness layer-c and variable stiffness layer-d as defined in Table 7 are utilized. Here the variable stiffness layer-d being different from the other layers
Table 6 Maximum transverse displacement and axial rotation of eccentrically loaded variable stiffness box-beam with varying fibre orientations. 00
150
300
450
600
750
900
0 150 300 450 600 750 900
0,317 0,321 0,333 0,343 0,338 0,307 h(s) ? 1
0,325 0,341 0,368 0,392 0,404 0,401 0,372
0,352 0,381 0,419 0,451 0,466 0,471 0,467
0,387 0,422 0,462 0,488 0,498 0,501 0,498
0,412 0,448 0,482 0,501 0,506 0,506 0,504
0,424 0,459 0,489 0,502 0,506 0,505 0,504
h(s) ? 1 0,46 0,488 0,501 0,504 0,504 0,504
(h1); (h0)?
00
150
300
450
600
750
900
10,17 8,34 6,09 4,69 3,83 3,36 h(s) ? 1
8,22 5,96 4,31 3,46 3,05 3,05 3,58
5,99 4,28 3,28 2,84 2,76 3,09 4,02
4,71 3,46 2,84 2,68 2,85 3,47 4,78
3,98 3,08 2,77 2,85 3,29 4,32 6,10
3,57 3,06 3,07 3,48 4,35 6,02 8,36
h(s) ? 1 3,40 3,87 4,75 6,18 8,46 10,32
(h1); (h0)? 0
V (mm)
/ 10
3
(rad)
0
0 150 300 450 600 750 900
Fig. 9. Contour plots of maximum transverse displacement and axial rotation of eccentrically loaded variable stiffness box-beam.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Fig. 10. Variable stiffness box-beam loaded at its free end.
Table 7 Definitions of variable stiffness layer-c and -d, the corresponding fibre paths and the thickness changes. Layer
h0
h1
h2
h0
href
c d
10 10
45 45
10 45
0,2 mm 0,2 mm
45 45
defined previously is asymmetric with different h0 and h2 values. This asymmetric layer is placed around box-beam by providing the orientation angle continuity at the corners of the beam. The numerical results obtained for different loading conditions and lay-ups of layer-c and -d are presented in Table 8 and compared with the results of a finite element software (FEAS), Abaqus, where 4-node shell elements are used. As it is seen in Table 8 the results presented are very close to the ones obtained by FEAS. It should be noted that for a further refer-
ence the displacements were also obtained for the beams with different lay-ups of layer-d in their vertical and horizontal walls. As it can be observed from numerical results tabulated in Table 8, W0 and U are coupled under the effect of the load Fz and moment Mz for both lay-ups of layer-c and -d. It can also be said that applying only the load Fy for the beam with lay-up of layer-c causes almost no coupling between V and other displacements while the use of the layer-d in beams creates a strong coupling between U and V displacements due to asymmetric definition of this layer.
Table 8 Free end displacements and axial rotation of the variable stiffness composite box-beam for different loading conditions. Lay-Up
Load
Results
W0 (mm)
U (mm)
V (mm)
U (rad)
Top, Bottom, Left, Right ½þ þ c =þ c 5
Fy = 1 kN, Mz = 1 kNm, Fz = 100 kN
Present FEAS Present FEAS Present FEAS Present FEAS
2,861 2,838 0 0 2,869 2,844 0,009 0,007
0 0 0 0 0 0 0 0
10,8 10,9 10,8 10,9 0 0 0 0
0,1044 0,1041 0 0 0,0008 0,0007 0,1052 0,1047
Present FEAS Present FEAS Present FEAS Present FEAS
4,672 4,626 0 0 4,684 4,632 0,012 0,006
24,5 23,7 24,5 23,7 0 0 0 0
29,7 29,2 29,7 29,2 0 0 0 0
0,0837 0,0837 0 0 0,0013 0,0007 0,0851 0,0844
Fy = 1 kN Fz = 100 kN Mz = 1 kNm
Top, Bottom, ½þ þ d=þ d5 Left, Right ½þ d= d5
Fy = 1 kN, Mz = 1 kNm, Fz = 100 kN Fy = 1 kN Fz = 100 kN Mz = 1 kNm
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
b66 ðsÞ ¼ B66 ðsÞ ½A22 ðsÞ B26 ðsÞ D26 ðsÞ B22 ðsÞfB26 ðsÞ2
6. Conclusion An analytical model was developed to investigate the flexuraltorsional behavior of single-cell closed cross-section thin -walled laminated composite beams with variable stiffness. Variable stiffness layers are acquired by using the curvilinear fibres. A procedure introduced for alignment and mapping of variable stiffness layers in a new scaled coordinate system facilitates to work with multiple variable stiffness layers. The symbolic notation presented makes it possible to show the variable stiffness layers in similar well-known lay-up notation used for classical constant stiffness layers. The numerical results of present model developed are initially obtained by use of a displacement based finite element method with 2-node beam elements for thin-walled having straight fibres and compared with the ones available in the literature After having seen the results agreed well with the ones of the previous constant stiffness case, new numerical results involving displacements and rotations for the same type of the beam with same conditions but with variable stiffness layers are also obtained and presented in order to see the effect of changing the orientation of fibre angle. The contour plots are presented to show that the same transverse displacement or axial rotation can be achieved for different fibre paths leading to a flexibility in the optimization process of such structures. Finally the composite box-beams made of two different fibre paths in their layers, clamped from one end and with four types of loading conditions are considered to investigate the resulting couplings between displacements. It is also observed that the numerical results obtained are in very good agreement with the results of the finite element analysis software. It can be said that using variable stiffness layers, that is, varying the paths of curvilinear fibres in structures provides more flexibility in their design to have a better mechanical performance in comparison with composite structures having classical straight fibres in their layers. Appendix A
þ A26 ðsÞ D26 ðsÞg þ A26 ðsÞ B26 ðsÞ D22 ðsÞ=D22 d11 ðsÞ ¼ D11 ðsÞ ½D12 ðsÞ2 A22 ðsÞ 2 D12 ðsÞ B12 ðsÞ B22 ðsÞ þ D22 ðsÞ B12 ðsÞ2 =D22 2 6 d16 ðsÞ ¼ D16 ðsÞ 4
2
þ A22 ðsÞ B12 ðsÞ =D22
ðA:1aÞ
þ B12 ðsÞ A26 ðsÞg þ A12 ðsÞ A26 ðsÞ D22 ðsÞ=D22
ðA:1bÞ
þ D22 ðsÞ B26 ðsÞ2 =D22
D22 ¼ A22 ðsÞ D22 ðsÞ B22 ðsÞ2
2
þ A22 ðsÞ B26 ðsÞ =D22
Z
l
k11 ¼ 0
Z
l
k12 ¼ 0
Z
l
k13 ¼ 0
Z
l
k14 ¼ 0
Z
l
k22 ¼
þ A12 ðsÞ D12 ðsÞg þ A22 ðsÞ B12 ðsÞ D12 ðsÞ=D22 6 a b16 ðsÞ ¼ B16 ðsÞ 4
A12 ðsÞ B26 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ B26 ðsÞ þA12 ðsÞ D26 ðsÞg
0
6 b b16 ðsÞ ¼ B16 ðsÞ 4
þA26 ðsÞ D12 ðsÞg
l
k33 ¼ 0 l
0
Z
l
0
ðA:1dÞ 3
Z f1 ¼
7 5=D22
3 7 5=D22
þA22 ðsÞ B26 ðsÞ D12 ðsÞ ðA:1fÞ
ðA:1jÞ ðA:2Þ
ðA:3aÞ
e12 S0L S00H
ðA:3bÞ
e13 S0L S00H
ðA:3cÞ
e15 S0L S0H e14 S0L S00H
ðA:3dÞ
e22 S00H S00H
ðA:3eÞ
e23 S00H S00H
ðA:3fÞ
e24 S00H S00H e25 S00H S0H
ðA:3gÞ
e33 S00H S00H
ðA:3hÞ
e34 S00H S00H e35 S00H S0H
ðA:3iÞ
e44 S00H S00H e45 ðS00H S0H þ S0H S00H Þ e55 S0H S0H
ðA:3jÞ
qz SL
ðA:4aÞ
qx SH
ðA:4bÞ
qy SH
ðA:4cÞ
mz SH
ðA:4dÞ
l
0
Z
e11 S0L S0L
l
0
Z f2 ¼
ðA:1eÞ A26 ðsÞ B12 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ B26 ðsÞ
l
k24 ¼
k44 ¼
þA22 ðsÞ B12 ðsÞ D26 ðsÞ 2
l
0
ðA:1cÞ
b11 ðsÞ ¼ B11 ðsÞ ½A12 ðsÞ B12 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ 2
Z k23 ¼
k34 ¼
2
7 5=D22
Elements of element stiffness matrix and element force vector in Eqs. (33) and (34).
Z
a66 ðsÞ ¼ A66 ðsÞ ½A26 ðsÞ2 D22 ðsÞ 2 A26 ðsÞ B26 ðsÞ B22 ðsÞ
þD12 ðsÞ B26 ðsÞg
3
d66 ðsÞ ¼ D66 ðsÞ ½D26 ðsÞ2 A22 ðsÞ 2 D26 ðsÞ B26 ðsÞ B22 ðsÞ
Z
a16 ðsÞ ¼ A16 ðsÞ ½A22 ðsÞ B12 ðsÞ B26 ðsÞ B22 ðsÞfA12 ðsÞ B26 ðsÞ
A22 ðsÞ D12 ðsÞ D26 ðsÞ B22 ðsÞfB12 ðsÞ D26 ðsÞ
ðA:1iÞ
Z
a11 ðsÞ ¼ A11 ðsÞ ½A12 ðsÞ2 D22 ðsÞ 2 A12 ðsÞ B12 ðsÞ B22 ðsÞ
ðA:1hÞ
þB12 ðsÞ B26 ðsÞ D22 ðsÞ
0
The detailed expressions for the modified stiffness coefficients in Eq. (23):
ðA:1gÞ
l
f3 ¼ 0
Z f4 ¼ 0
l
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References [1] Vlasov VZ. Thin walled elastic beams. 2nd ed. Jerusalem: Israel Program for Scientific Transactions; 1961. [2] Gjelsvik A. The theory of thin-walled bars. New York: Wiley; 1981. [3] Chandra R, Stemple AD, Chopra I. Thin-walled composite beams under bending, torsional and extensional loads. J Aircraft 1990;28(7):619–36. [4] Song O, Librescu L. Free vibration of anisotropic composite thin-walled beams of closed cross-section contour. J Sound Vib 1993;167(1):129–47. [5] Kollar LP, Pluzsik A. Analysis of thin-walled composite beams with arbitrary layup. J Reinf Plast Compos 2002;21(16):1423–65. [6] Piovan TM, Cortinez VH. Mechanics of shear deformable thin-walled beam made of composite materials. Thin-Walled Struct 2007;45:37–62. [7] Vo TP, Lee J. Flexural–torsional behavior of thin-walled closed-section composite box beams. Eng Struct 2007;29:1774–82. [8] Vo TP, Lee J. Geometrically nonlinear analysis of thin-walled composite box beams. Comput Struct 2009:236–45. [9] Sheikh AH, Thomsen OT. An efficient beam element for the analysis of laminated composite beams of thin walled open and closed cross sections. Compos Sci Technol 2008;68(10):2273–81. [10] Zhang C, Wang S. Structure mechanical modeling of thin-walled closed-section composite beams, Part1: single-cell cross-section. Compos Struct 2014;113:12–22. [11] Carrera E, Filippi M, Mahato PK, Pagani A. Accurate static response of singleand multi-cell laminated box beams. Compos Struct 2016;136:372–83. [12] Martin AF, Leissa AW. Application of the Ritz method to plane elasticity problems for composite sheets with variable fibre spacing. Int J Numer Methods Eng 1989;28:1813–25.
[13] Gürdal Z, Olmedo R. In-plane response of laminates with spatially varying fibre orientations: variable stiffness concept. AIAA J 1993;31(4):751–8. [14] Gürdal Z, Tatting BF, Wu CK. Variable stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos Part A 2008;39(5):911–22. [15] Akhavan H, Ribeiro P, de Moura MFSF. Large deflection and stresses in variable stiffness composite laminates with curvilinear fibres. Int J Mech Sci 2013;73:14–26. [16] Zamani Z, Haddadpour H, Ghazavi MR. Curvilinear fibre optimization tools for design thin walled beams. Thin-Walled Struct 2011;49(3):448–54. [17] Librescu L, Song O. Thin-walled composite beams. Springer; 2006. p. 7–32. [18] Smith C, Chopra I. Formulation and evaluation of an analytical model for composite box-beams. J Am Helicopter Soc 1990;36(3):23–35. [19] Kim BC, Potter K, Weaver PM. Continuous tow shearing for manufacturing variable angle tow composites. Compos Part A 2012;43:1347–56. [20] Kim BC, Potter K, Weaver PM. Manufacturing characteristics of the continuous tow shearing method for manufacturing of variable angle tow composites. Compos Part A 2014;61:141–51. [21] Jones RM. Mechanics of composite materials. New York: Hemisphere Publishing Corp; 1975. p. 187–201. [22] Kollar LP, Springer GS. Mechanics of composite structures. 1st ed. Cambridge University Press; 2003. [23] Kim C, White SR. Thick-walled composite beam theory including 3-D elastic effects and torsional warping. Int J Solid Struct 1997;34(31–32):4237–59. [24] Qin Z, Librescu L. On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations. Compos Struct 2002;56:345–58.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Static analysis of thin-walled laminated composite closed-section beams with variable stiffness M. Gökhan Günay, Taner Timarci ⇑ Trakya University, Mechanical Engineering Department, Edirne 22180, Turkey
a r t i c l e
i n f o
Article history: Received 4 July 2017 Revised 15 August 2017 Accepted 29 August 2017 Available online 1 September 2017 Keywords: Composite materials Thin-walled beams Variable stiffness Finite element analysis
a b s t r a c t Static behavior of thin-walled laminated composite closed cross-section beams having variable stiffness is investigated in this study. The analytical model used accounts for flexural–torsional coupling and warping effects as well as the variable stiffness along the contour of the cross-section of the beam. The variable stiffness is acquired by constructing the laminates with curvilinear fibres having certain specific paths. The orientation of fibres varies by depending on the fibre path along the contour of the crosssection in each layer. Equilibrium equations are derived by use of minimum potential energy principle. Although the formulation given can be applied to any shape of the closed cross-section with straight or curved edges, preliminary numerical results are presented only for box-beams. A displacement based finite element method is developed to solve the analytical model and to predict displacements and rotations under the effect of different types of loading conditions. Numerical results are obtained for different fibre paths and lay-up configurations and compared with the available solutions in the literature also with the results of a finite element analysis software using shell element. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Thin walled composite laminated beams are widely used as structural elements in many engineering applications. The structures made of composite materials have many advantages such as having the high specific modulus and strength compared to the structures made of conventional materials. One other important advantage of composite layered structures reinforced with continuous fibres is that they can be tailored according to the requirements of their application area by changing, for example, thicknesses and the stacking sequences of layers also the orientation of fibres in layers. The theoretical background of thin-walled members are mostly based on the works of Vlasov [1] and Gjelsvik [2] where the beams are made of isotropic materials. There are many researches realised for the understanding the mechanical behavior of thin-walled beams made of composite materials. The coupling effects for symmetric and anti-symmetric box-beams under flexural, torsional, and extensional loads are presented in the work of Chandra et al. [3]. Song and Librescu [4] formulated and investigated the free vibration behavior of laminated composite thick and thin walled, single-cell beams of arbitrary crosssections. A beam theory including expressions for the stiffness ⇑ Corresponding author. E-mail address: [email protected] (T. Timarci). http://dx.doi.org/10.1016/j.compstruct.2017.08.092 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved.
matrix for thin-walled open and closed-section composite beams with arbitrary lay-ups was developed by Kollar and Pluzsik [5]. Piovan and Cortinez [6] presented a theoretical model for the dynamic, static and buckling analyses of closed cross-section composite thin-walled beam by taking into account the shear flexibility due to bending and warping. Vo and Lee [7] developed a general analytical model for flexural-torsional behavior of thin-walled composite box-beams. The same authors extended their approach for the geometric nonlinearity [8]. Sheikh and Thomson [9] adopted a closed-form approach to derive the cross-sectional stiffness matrix for open and closed-section composite beams taking into account the coupling effects and used a three-node beam element in the finite element analysis. a one-dimensional beam model considering the effects of axial warping and material anisotropies for single-cell composite beams was developed by Zhang and Wang [10]. Carrera et al. [11] realised the static analysis of laminated beams with thin-walled cross-sections in the frame of Carrera Unified Formulation (CUF) based on Taylor (TE) and Lagrange Expansions (LE) of cross-sectional functions, leading to accurate and computationally efficient finite element methods. Composite structures are generally produced by using straight and evenly distributed fibres. An additional useful property can be added to the composite laminated structures by using curvilinear fibres in layers instead of using conventional straight fibres. This provides another flexibility in design of such structures with
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
variable stiffness laminates to satisfy certain mechanical requirements. The mechanical properties of structural elements can also be varied by changing the distribution or the shape of fibres. Martin and Leissa [12] played with the distance between fibres to achieve a variable stiffness composite plate and applied Ritz method to several plane elasticity problems. The curved fibres were used by Gürdal and Olmedo [13] to vary stiffnesses of rectangular composite plates. They analyzed the elastic response of the variable stiffness panel by use of closed-form and numerical solutions. It was also shown by Gürdal et al. [14] that it is possible to change either the buckling load or the in-plane stiffness while keeping the other one constant for variable stiffness panels. Variable stiffness concept is investigated commonly on plates and cylinders but rarely on thin-walled beams. In the paper of Akhavan at al. [15] large deflections and stresses of variable stiffness composite laminated plates with curvilinear fibres were calculated and it was shown that the fibre variation may be used to reduce deflections and stresses in some static loadings. Zamani et al. [16] used curvilinear fibres to optimize thin-walled beams with bi-convex cross section. In the present study the static behavior of thin-walled composite beams with variable stiffness and thickness is investigated. The formulation developed for thin-walled composite beams with closed cross-section is based on the theory given in Librescu and Song [17]. On the basis of this formulation the present model accounts for the variable stiffness as well as the variation of thickness in layers. A mapping procedure to facilitate the usage of different fibre angle variations in layers is proposed, furthermore a new description and lay-up code is also used to define variable stiffness layers on thin-walled composite beams in a similar manner of straight fibres case. Although the formulation presented can be applied to any shape of the closed cross-section with straight or curved edges, preliminary numerical results are presented only for the composite box-beams. A displacement based finite element method which is parallel to the presented in Vo and Lee [7] is used to solve the analytical model and to predict displacements and rotations under the effect of different types of loading conditions. Numerical results obtained for different fibre paths and lay-up configurations are presented and compared with the results of a commercial finite element analysis software in which the shell element is utilized. 2. Kinematics The two coordinate systems used are shown in Fig. 1. The first coordinate system is the global cartesian (x, y, z) coordinate system. The x- and y-axes lie in the plane of the cross-section and the z-axis is along the longitudinal axis of the beam. The second one is the local coordinate system (n, s, z) where n-axis is normal
to the middle surface as s-axis is tangent to the middle surface and it is directed along the contour line of the cross-section. These two coordinate systems are related through an angle of orientation a. Point P called as the pole point is used as a reference to define displacements and rotations of cross section. U and V are displacements of the pole point in the global coordinate system along x and y axes. / is the rotation of cross section around the pole point. 2.1. Assumptions The following assumptions are made in developing the kinematical relations. (i) The cross section of beam is assumed to maintain its shape in its own plane during deformation. There is no cross sectional distortion although it is allowed to warp out of its own plane. (ii) The normals to the undeformed middle surface remain normal to the deformed middle surface. (iii) The ratio of the wall thickness to the radius of curvature at any point of the beam wall is negligibly small compared to unity. (iv) The hoop stress rs in the contour direction s is small compared to the axial stress rz . (v) Transverse shear strains cxz and cyz are neglected. 2.2. Displacement field Displacements of an arbitrary point m(x,y) on the middle surface can be expressed by the displacements of point P(xp, yp) in the local coordinate system as given below [16]:
n ðs; zÞ ¼ UðzÞ u
dy dx VðzÞ /ðzÞ r s ds ds
ð1aÞ
s ðs; zÞ ¼ UðzÞ u
dx dy þ VðzÞ þ /ðzÞ r n ds ds
ð1bÞ
n; zÞ ¼ W 0 ðzÞ U 0 ðzÞ xðsÞ V 0 ðzÞ yðsÞ /0 ðzÞ wðsÞ wðs;
ð1cÞ
where W 0 is average axial displacement of cross section, wðsÞ is primary warping function. r n ðsÞ is the normal component and r s ðsÞ is the tangential component of the distance between points P and m as described below:
rn ðsÞ ¼ ðxðsÞ xp Þ
dy dx ðyðsÞ yp Þ ds ds
ð2aÞ
rs ðsÞ ¼ ðxðsÞ xp Þ
dx dy þ ðyðsÞ yp Þ ds ds
ð2bÞ
Similarly displacements of an arbitrary point M (X,Y) laying out of the middle surface can be defined in the local coordinate system as:
U n ðs; n; zÞ ¼ UðzÞ
dy dx VðzÞ /ðzÞ Rs ðs; nÞ ds ds
ð3aÞ
U s ðs; n; zÞ ¼ UðzÞ
dx dy þ VðzÞ þ /ðzÞ Rn ðs; nÞ ds ds
ð3bÞ
dy dx V 0 ðzÞ yðsÞ n Wðs; n; zÞ ¼ W 0 U 0 ðzÞ xðsÞ þ n ds ds þ n wðsÞ /0 ðzÞ ½wðsÞ
Fig. 1. Displacements of an arbitrary point (m) on the middle surface.
ð3cÞ
where wðsÞ is the secondary warping function, Rn ðs; nÞ is the normal component and Rs ðs; nÞ is the tangential component of the distance
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
where primes ()’ denote the derivative with respect to z. ezz and csz strains can be separated into to middle surface strains ()(0) and middle surface curvatures ()(1) as given below: ð1Þ ezz ¼ eð0Þ zz þ n ezz
ð9aÞ
ezzð0Þ ¼ w00 U00 x V00 y /00 w
ð9bÞ
00 eð1Þ zz ¼ U
dy dx þ V00 /00 wðsÞ ds ds
ð1Þ csz ¼ cð0Þ sz þ n csz
cszð0Þ ¼
Fig. 2. An arbitrary point (M) out of the middle surface.
ð9cÞ ð10aÞ
2X ; hðsÞ Gsz ðsÞ L
cszð1Þ ¼
2b hðsÞ Gsz ðsÞ L
ð10bÞ
between points P and M as shown in Fig. 2. They can be expressed by the terms rn ðsÞ and r s ðsÞ as:
2.4. Governing equations
Rn ðs; nÞ ¼ r n ðsÞ þ n
ð4aÞ
Rs ðs; nÞ ¼ r s
ð4bÞ
Equilibrium equations of the thin-walled beam can be obtained by equating the variation of the total potential energy of the system to zero. The total potential energy of the system, Ptotal , is the sum of strain energy, Pstrain , and work done by external forces and moments, W ext as, given below:
Definitions of the primary and secondary warping functions are given below:
Z s wðsÞ ¼ rn 0
Z s wðsÞ ¼
2X ds hðsÞ Gsz ðsÞ L
ð5aÞ
2X ¼
I
I
r
ðsÞ ds;
b¼
I
d
L¼
s;
ds hðsÞGsz ðsÞ
Z 1 ðrðkÞ ezz þ rðkÞ sz csz Þdv 2 v zz
ð5bÞ
where hðsÞ is the thickness of the cross-section, Gsz ðsÞ is the effective shear stiffness of the laminate. 2X, b being the double area enclosed by the center line of the beam cross-section and its perimeter, respectively and L are defined as follows:
ð11Þ
The strain energy is:
Pstrain ¼
b ds rs hðsÞ Gsz ðsÞ L
Ptotal ¼ Pstrain þ W ext
By substituting Eqs. (9) and (10) into Eq. (12) the strain energy can be obtained as:
Z 1 dy dx V 00 ðy n Þ rzzðkÞ ðsÞ W 0 0 U 00 x þ n 2 v ds ds 1 2 X 2b /00 wðs; nÞ þ rðkÞ /0 dv þn sz ðsÞ hðsÞ Gsz ðsÞ L L
Pstrain ¼
ð6Þ
ð13Þ
n
For a thin-walled laminated composite beam the effective shear stiffness Gsz ðsÞ given in [18] can be calculated as expressed below:
"
2
1 a16 ðsÞ Gsz ðsÞ ¼ a66 ðsÞ hðsÞ a11 ðsÞ
#
ð7Þ
where aij ðsÞ are modified beam stiffnesses which are defined explicitly in Appendix A.
This equation can be rearranged as:
ezz
dy dx ¼ W 0 0 U 00 x þ n V 00 y n /00 wðs; nÞ ds ds
0
l
ðNz W 00 M x V00 M y U00 þ M z /0 M x /00 Þdz ð14Þ
where the beam forces and the moments are defined as:
Z
Nz ¼
Nzz ðsÞ ds
ð15aÞ
Z Mx ¼
Z
ð8eÞ
x Nzz ðsÞ Lzz ðsÞ
dy ds ds
ð15cÞ
ðsÞ Lzz ðsÞ xðsÞds Nzz ðsÞ x
ð15dÞ
c
Z Mz ¼
ð8dÞ
ð15bÞ
c
ð8bÞ
2X 2b csz ¼ þn hðsÞ Gsz ðsÞ L hðsÞ Gsz ðsÞ L
dx ds ds
Z My ¼ Mx ¼
ð8cÞ
y Nzz ðsÞ þ Lzz ðsÞ c
ð8aÞ
@W @U s þ csz ¼ ds @z
@W @U n þ cnz ¼ ¼0 @n @z
Z
c
By the assumption (i) in-plane strains will be zero (exx ¼ 0, eyy ¼ 0, exy ¼ 0). Axial strain ezz and shear strains csz and cnz can be calculated with small displacements assumption as:
ezz
1 2
Pstrain ¼
2.3. Strain field
dW ¼ dz
ð12Þ
Nsz ðsÞ c
2X 2b Lsz ðsÞ ds hðsÞ Gsz ðsÞ L hðsÞ Gsz ðsÞ L
ð15eÞ
Here N ab ðsÞ and Lab ðsÞ (as a,b = s,z) are shell stress resultants are couples along the thickness of the laminate respectively as given below and as shown in Fig. 3.
Nab ðsÞ ¼
N Z X k
n;k
n;k1
rðkÞ ab dn
ð16aÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Fig. 3. (a) Shell stress resultants and (b) couples.
Lab ðsÞ ¼
N Z X k
n;k
n;k1
ðkÞ
n rab dn
ð16bÞ
The work done by external forces and moments is given as:
Z W ext ¼ 0
l
ðqz W 0 þ qx U þ qy V þ mz /Þdz
ð17Þ
where qx and qy are transverse loads, qz is axial load and mz is axial torque. By equating the variation of the total potential energy to zero, the following weak formulation can be obtained:
dPtotal ¼ dPstrain þ dW ext ¼ 0 Z 0¼ 0
l
nesses. Calculations of these variable stiffness elements are explained in next section. Based on the assumption (iv) transverse stress resultant N ss ðsÞ and transverse stress couple Lss ðsÞ will be very small compared to other stress resultants and couples. By taking these hoop terms as zero as below:
essð0Þ ¼
þ qz dW 0 þ qx dU þ qy dV þ mz d/Þdz
þ ðA26 ðsÞ D22 ðsÞ B22 ðsÞ B26 ðsÞÞcð0Þ zz þ ðB26 ðsÞ D22 ðsÞ ð1Þ B22 ðsÞ D26 ðsÞÞczz
eð1Þ ss ¼ ð19Þ
ð20aÞ
M 00y þ qx ¼ 0
ð20bÞ
M 00x þ qy ¼ 0
ð20cÞ
M 00w þ M0z þ mz ¼ 0
ð20dÞ
2.5. Constitutive equations The constitutive relations between shell stress resultants and couples and strains of a composite shell consisting of the layers with variable stiffness are defined as follows:
3 2 3 A11 ðsÞ A12 ðsÞ A16 ðsÞ B11 ðsÞ B12 ðsÞ B16 ðsÞ N zz ðsÞ 6 N ðsÞ 7 6 A ðsÞ A ðsÞ A ðsÞ B ðsÞ B ðsÞ B ðsÞ 7 22 26 12 22 26 7 6 ss 7 6 12 7 6 7 6 6 N sz ðsÞ 7 6 A16 ðsÞ A26 ðsÞ A66 ðsÞ B16 ðsÞ B26 ðsÞ B66 ðsÞ 7 7 6 7 6 6 L ðsÞ 7 ¼ 6 B ðsÞ B ðsÞ B ðsÞ D ðsÞ D ðsÞ D ðsÞ 7 12 16 11 12 16 7 6 11 7 6 zz 7 6 7 6 4 Lss ðsÞ 5 4 B12 ðsÞ B22 ðsÞ B26 ðsÞ D12 ðsÞ D22 ðsÞ D26 ðsÞ 5 2
Lsz ðsÞ
B16 ðsÞ B26 ðsÞ 2 ð0Þ 3
B66 ðsÞ
D16 ðsÞ D26 ðsÞ D66 ðsÞ
ezz
1 A22 ðsÞ D22ðsÞ B222 ðsÞ
ð22aÞ ½ðA12 ðsÞ B22 ðsÞ B12 ðsÞ A22 ðsÞÞeð0Þ zz
þ ðB12 ðsÞ B22 ðsÞ A22 ðsÞ D12 ðsÞÞeð1Þ zz þ ðA26 ðsÞ B22 ðsÞ A22 ðsÞ B26 ðsÞÞcð0Þ zz þ ðB26 ðsÞ B22 ðsÞ
Equilibrium equations of beam can be obtained by integrating derivatives of varied quantities by parts in Eq. (19) and collecting the coefficients of dw0 , dU, dV and d/ as:
N0z þ qz ¼ 0
½ðA12 ðsÞ D22 ðsÞ B12 ðsÞ B22 ðsÞÞeð0Þ zz
þ ðB12 ðsÞ D22 ðsÞ B22 ðsÞ D12 ðsÞÞeð1Þ zz
ð18Þ
ðNz dW 00 Mx dV 00 M y dU 00 þ M z d/0 Mx d/00
1 A22 ðsÞ D22ðsÞ B222 ðsÞ
ð1Þ eð0Þ ss and ess strains can be expressed
A22 ðsÞ D26 ðsÞÞcð1Þ zz
ð22bÞ
By substituting Eqs. (22) into Eq. (21) the modified constitutive relations can be expressed as given below:
3 2 a ðsÞ a ðsÞ b ðsÞ 11 16 11 Nzz ðsÞ b 6 N ðsÞ 7 6 ðsÞ a ðsÞ b a 16 66 6 sz 7 6 16 ðsÞ 7¼6 6 6 4 Lzz ðsÞ 5 4 b ðsÞ bb ðsÞ d ðsÞ 11 11 16 a Lsz ðsÞ b ðsÞ b66 ðsÞ d16 ðsÞ 2
16
a
b16 ðsÞ
3 2
3
eð0Þ zz 7 6 ð0Þ 7 6 csz 7 b66 ðsÞ 7 76 7 7 6 ð1Þ 7 d16 ðsÞ 5 4 ezz 5 cð1Þ d66 ðsÞ sz
ð23Þ
The detailed expressions for the stiffness coefficients aij ðsÞ bij ðsÞ and dij ðsÞ are given in Appendix A. The constitutive relations between the beam forces and moments and the derivatives of the pole point displacement terms can be obtained by substituting modified constitutive relations into Eqs. (15) as:
3 2 e11 Nz 7 6 6 6 My 7 6 7 6 6 6 Mx 7 ¼ 6 7 6 6 7 6 6 4 Mx 5 4 Mz 2
e12
e13
e14
e22
e23
e24
e33 sym
e34 e44
3 W 00 7 6 7 e25 7 6 U 00 7 7 6 7 00 7 6 e35 7 7 6 V 7 7 6 7 e45 5 4 /00 5 0 e55 / e15
3 2
ð24Þ
where ½eij is beam stiffness matrix of thin-walled composite laminated beam with variable stiffness. Elements of ½eij can be expressed as:
Z
6 ð0Þ 7 6 ess 7 7 6 6 cð0Þ 7 6 sz 7 6 ð1Þ 7 6 ezz 7 7 6 6 ð1Þ 7 4 ess 5
e11 ¼
a11 ðsÞ ds
ð25aÞ
s
e12
cszð1Þ ð21Þ Here, the elements of A, B, D matrices are the functions of s along the beam cross-sectional contour which differ from the conventional definitions of extensional, coupling and bending stiff-
e13
Z dy ds ¼ a11 ðsÞ xðsÞ þ b11 ðsÞ ds s Z dx ds ¼ a11 ðsÞ yðsÞ b11 ðsÞ ds s Z
e14 ¼ s
1 2X 2b a ds a16 ðsÞ þ b16 ðsÞ hðsÞ Gsz ðsÞ L L
ð25bÞ ð25cÞ
ð25dÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Z e15 ¼
ðsÞ þ b11 ðsÞ xðsÞÞ ds ða11 ðsÞ x
ð25eÞ
iv iv iv III e13 W III 0 e23 U e33 V e34 / þ e35 / þ qy ¼ 0
ð26cÞ
s
Z e22 ¼ s
2 ! dy dy þ d11 ðsÞ a11 ðsÞ xðsÞ þ 2 b11 ðsÞ xðsÞ ds ds ds ð25fÞ
e23
Z dy dx ¼ ða11 ðsÞ xðsÞ yðsÞ þ b11 ðsÞ yðsÞ xðsÞ ds ds s dx dy ds d11 ðsÞ ds ds 1 2X 2b a a16 ðsÞ xðsÞ þ b16 ðsÞ xðsÞ L L s hðsÞ Gsz ðsÞ 2X dy 2b dy b þ d16 ðsÞ ds þ b16 ðsÞ L ds L ds
ð25gÞ
Z
e24 ¼
e25 ¼
ð25hÞ
Z dy ðsÞ þ b11 ðsÞ ðx ðsÞ þ xðsÞ xðsÞÞ a11 ðsÞ xðsÞ x ds s dy ds ð25iÞ þ d11 ðsÞ xðsÞ ds Z
e33 ¼ s
2 ! dx dx 2 ds a11 ðsÞ yðsÞ 2 b11 ðsÞ yðsÞ þ d11 ðsÞ ds ds ð25jÞ
1 2X 2b a a16 ðsÞ yðsÞ þ b16 ðsÞ yðsÞ L L s hðsÞ Gsz ðsÞ 2X dx 2b dx b ds b16 ðsÞ d16 ðsÞ L ds L ds Z
e34 ¼
e35 ¼
ð25kÞ
Z dx ðsÞ b11 ðsÞ ðx ðsÞ yðsÞ xðsÞÞ a11 ðsÞ yðsÞ x ds s dx ds ð25lÞ d11 ðsÞ xðsÞ ds 2X 2X 2X 2b a66 ðsÞ þ 2 b66 ðsÞ 2 2 L L L L s hðsÞ Gsz ðsÞ 2b 2b ds ð25mÞ þ d66 ðsÞ L L Z
e44 ¼
1
1 2X 2b a ðsÞ ðsÞ a16 ðsÞ x þ b16 ðsÞ x L L s hðsÞ Gsz ðsÞ 2X 2b b ds þ d16 ðsÞ xðsÞ þb16 ðsÞ xðsÞ L L Z
e45 ¼
e55 ¼
ð25nÞ
Z ðsÞ2 þ b11 ðsÞ x ðsÞ xðsÞ þ d11 ðsÞ xðsÞ2 ds a11 ðsÞ x s
ð25oÞ Diagonal elements of ½eij are primary stiffnesses related to the extension along z-axis, to the bending about y- and x-axes and to the bi-moment and torsion about z-axis. Remaining elements of ½eij are the stiffnesses of the coupled behaviors. By substituting Eq. (24) into Eqs. (20) the explicit form of the governing equations are obtained in terms of these beam stiffnesses and the derivatives of pole displacements as:
e11
W II0
e12
W III 0
II iv III iv III e14 W III 0 þ e15 W 0 e24 U e25 U e34 V e35 V
e44 /iv þ 2e45 /III þ e55 /II þ mz ¼ 0
2
e12 U
IIIe13
iv
III
III
II
V e14 / þ e15 / þ qz ¼ 0 iv
iv
III
e22 U e23 V e24 / þ e25 / þ qx ¼ 0
ð26aÞ ð26bÞ
ð26dÞ
3. Description of the variable stiffness Stiffnesses of a continuous fibre reinforced composite layer depends on the orientation angle of the fibres. A desired orientation angle variation can be achieved by placing curvilinear fibres to follow a specific path. Thus stiffness of a layer produced with curvilinear fibres will be spatially varying depended on the position and the direction. A simple procedure is applied in order to facilitate in handling beams composed of variable stiffness layers with different configurations. In this procedure the variable stiffness layers are defined on a new f g plane separately corresponding to the local s-z plane and a name is given to each layer. Fibre orientation angle variation of a variable stiffness layer is expressed in terms of these coordinates same as the one presented in the work of Gürdal and Olmedo [13] formulation by using a linear interpolation as:
hðfÞ ¼
ðh1 h0 Þ f þ h1 ; 1 6 f 6 0 ðh2 h1 Þ f þ h1 ; 0 < f 6 1
ð27Þ
where h1 expresses the orientation angle at the origin, h0 and h2 express the orientation angles at f ¼ 1 and f ¼ 1 respectively. This orientation angle variation is satisfied by the following the fibre path definition: 8 ln½cosððh0 h1 Þ f þ 90 h1 Þ þ ln½cosð90 h1 Þ > > ; 1 6 f 6 0 > h1 Þ < p ðh0180 gðfÞ ¼ > ln½cosððh1 h2 Þ f þ 90 h1 Þ þ ln½cosð90 h1 Þ > > ; 0
ð28Þ As an example, a variable stiffness layer named as ‘‘a” is defined in Table 1. Fibre orientation angle variation, the corresponding fibre path and the change of the thickness is also shown in Fig. 4. A layer with variable stiffness defined in such a way can be mapped on a layer of any desired segment of the thin-walled composite beam as shown in Fig. 5. In Table 2 it is seen that f g alignments can be done in four different ways by flipping f g plane. The mapping of the variable stiffness layer- a on the kth layer of segment-i is given for each alignment where si1 and si denote the borders of the ith segment. Symbolic representation of the alignments for layer-a are expressed by ðfÞ ðgÞ a, where (+) sign means the default direction and () means the flipped direction of the corresponding axes. It is assumed that the variable stiffness layers are produced by shifting the fibre path along g axis and placing the new fibres next to the previous ones such that any gap or overlap among layers is not allowed to occur. As a result, the stiffness of the layer will be constant along g axis while will be varying along f axis. An experimental production method called as Continuous Tow Shearing [19,20] developed recently offers the capability to fulfilling these requirements while the other methods such as Automated and Tailored Fibre Placement cause the gaps or overlaps in the structure. On the
Table 1 Definition of variable stiffness layer-a. Layer
h0
h1
h2
h0
href
a
700
200
700
1 mm
200
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Fig. 4. (a) Orientation angle variation, (b) corresponding fibre path and (c) thickness change of variable stiffness layer-a.
Fig. 5. Mapping of variable stiffness layer on the first segment of the thin-walled composite beam.
other hand in Continuous Tow Shearing method the thickness of the produced layer varies by depending on the change of the orientation angle. This relation can be expressed as follows:
here nk ðsÞ terms will depend on the thickness change of layers in the lay-up.
hðfÞ ¼ h0 =cosðhðfÞ href Þ
4. Finite element method
ð29Þ
Here href is the reference angle of the region where the thickness of the layer is equal to the original thickness (ho ). Upon mapping the elements of A(s), B(s), D(s) matrices are calculated by using the well known formulations [21] depending on hðsÞk as:
Aij ðsÞ ¼
N X ij ðhðsÞ Þ:ðnk ðsÞ nk1 ðsÞÞ Q k
ð30aÞ
k¼1
The governing equations are solved with displacement based finite element method by following similar procedure presented in [7]. 2-node beam elements are used to discretize thin-walled beam. Each node has seven degrees of freedom; three for translation, three for rotation and one for the rate of twist. Generalized displacements are expressed as:
Wo ¼
N X wi SL;i
ð31aÞ
i¼1
Bij ðsÞ ¼
Dij ðsÞ ¼
N 1X ij ðhðsÞ Þ:ðn2 ðsÞ n2 ðsÞÞ Q k k1 k 2 k¼1
ð30bÞ
N 1X ij ðhðsÞ Þ:ðn3 ðsÞ n3 ðsÞÞ Q k k1 k 3 k¼1
ð30cÞ
U¼
N X ui SH;i
ð31bÞ
i¼1
V¼
N X
v i SH;i
i¼1
ð31cÞ
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78 Table 2 Alignment, symbolic representation and formulation of mapping variable stiffness layer-a to the kth layer of ith segment where 1 f 1 si1 s si .
U¼
N X /i SH;i
ð31dÞ
i¼1
½K fDg ¼ fFg
ð32Þ
The global stiffness matrix ½K and force vector fFg can be constructed by summing the associated terms of the beam elements sharing the same node and by using the element stiffness matrix ½k and element force vector ff g which are given as:
6 6 ½k ¼ 6 4
hðsÞk ¼ hðfÞa
þ a
hðsÞk ¼ hðfÞa
þa
hðsÞk ¼ hðfÞa
a
hðsÞk ¼ hðfÞa
5. Numerical examples 5.1. Straight fibres case
where SL;i is Lagrange linear shape function and SH;i is Hermite-cubic shape function associated with ith node. By substituting these expressions into Eq. (19) the finite element formulation is obtained in the matrix form as given below where ½K is global stiffness matrix, fFg is nodal force vector and fDg is nodal displacement vector:
2
þ þa
k11
k12
k13
k22
k23
k14
k24 7 7 7 k34 5
k33 sym:
ff g ¼ f f 1
3
ð33Þ
k44 f2
f3
T
f4 g
ð34Þ
Elements of ½k and ff g are given in explicitly in Appendix A. After applying the boundary conditions and the external forces, unknown nodal displacements are obtained by solving Eq. (32).
Validation of the present formulation is firstly realised by comparing the results obtained with the ones of previous works in which the conventional straight fibres in the layers of composite box-beam are used. The two cases of composite box-beams having material properties given in Table 3 and the geometry and the loading conditions shown in Figs. 6 and 7 same with Refs. [3,11,23,24] and [8,9,22] are considered in the numerical results obtained. Initially, for the case of Fig. 6 with Material-1, variations of the angle of twist along the axis of the beam subjected to 0.113 N m tip torque, with 6-layers for two circumferentially uniform stiffness (CUS2 and CUS3) lay-ups are obtained and compared with the experimental results of Chandra et al. [3] and the ones obtained by Kim and White [23] Qin and Librescu [24] and by both LE and TE models of Carrera et al. [11], as shown in Fig. 8. The beam CUS2 has a [00/300]3 stacking sequence in the top flange and the left web and [00/300]3 sequence in the bottom flange and the right web. For the lay-up CUS3, the only difference is that 450 fibre orientation angle in layers replaces 300 angle of CUS2. It can be said that the results of the present approach are in good agreement with the ones of the previous works as they are in between of the solutions of two models of CUF presented by Carrera et al.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
[8,9,22]. As seen from the table the results obtained are very close the ones of the previous works for the plane strain case.
Table 3 Composite material properties. Material-1 [3,11,23,24] Material-2 [8,9,22]
E11 = 142 GPa
E22 = 9,79 GPa
G12 = 6 GPa
v12 = 0,42
E11 = 148 GPa
E22 = 9,65 GPa
G12 = 4,55 GPa
v12 = 0,34
5.2. Curvilinear fibres case
[11] for CUS3 and closer to the ones of Refs. [23,24] for CUS2. For the case of the composite box-beam shown in Fig. 7 with Material-2, the beam is clamped at each end and loaded with 6,5 kN/m eccentric distributed load. Lay-up of each wall of the beam is [[45/45]2/012/[45/45]2] with total layer thickness of 2 mm. The results for maximum transverse displacement and axial rotation are given in Table 4 and compared with the ones of Refs.
As the main objective of the present work, the thin-walled boxbeam shown in Fig. 8 with the same geometry and loading and boundary conditions is considered for the variable stiffness case, again with Material-2 properties. The new lay-up of the beam is þ represented as ½ðþ þ b=þ bÞ2 =012 =ðþ b=þ bÞ2 and layer-b is defined in Table 5. Angles of h0 and h1 are changed from 00 to 900 by 150 and the values of maximum transverse displacement and axial rotation are obtained for each combination and presented in Table 6 where hh0 ¼ h1 ¼ 450 i case corresponds to the previous example of thin-walled beam with straight fibres. The thickness
Fig. 6. Clamped-free box-beam with a torque subjected at its tip.
Fig. 7. Eccentrically uniform distributed loaded box-beam.
Fig. 8. Angle of twist distribution along the cantilever box-beam.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78 Table 4 Maximum transverse displacement and maximum axial rotation of eccentrically loaded composite box-beam. Reference
V (mm)
U 103 (rad)
Sheikh & Thomsen (rs = 0) [9] Sheikh & Thomsen (es = 0) [9] Vo & Lee (rs = 0) [8] Vo & Lee (es = 0) [8] Kollar & Springer [22] Present
0,494 0,438 0,494 0,438 0,488 0,488
6,70 2,76 6,43 2,68 2,76 2,68
Table 5 Definition of variable stiffness layer-b. Layer
h0
h1
h2
h0
href
b
0:15:90
0:15:90
h2 ¼ h0
0,1 mm
href ¼ h1
of the layer goes to infinity as the difference between h0 and h1 reaches to 900. Therefore this kind of combinations are not taken into consideration. It is seen that in cases of hh0 ¼ h1 ¼ 00 i and hh0 ¼ 00 ; h1 ¼ 750 i the transverse displacements are smallest
when hh0 ¼ h1 ¼ 00 i and hh0 ¼ 00 ; h1 ¼ 750 i combinations lead to a higher and a smaller value for the axial rotation, respectively. It can be said that it is possible to have a better structural performance as required by designing laminated structures with variable stiffness providing more alternatives in their design compared to the ones with conventional straight fibres. For the same beam, the maximum transverse displacement and maximum axial rotation results of the variable stiffness beam by varying h0 and h1 values are obtained and given in Fig. 9 as contour plots showing the curves of the equivalent values for the displacements and rotations scaled by the values for hh0 ¼ h1 ¼ 450 i case of straight fibres. In these plots it is seen that same transverse displacement or axial rotation can be achieved with various values of h0 and h1 which leads to have a flexibility for the optimization process in designing the structure. As another application of the present formulation, the composite box-beams with variable stiffness, clamped from one end with different types of loading as axial Fz, transverse Fy and Fz as well as the torsional moment Mz applied separately and as their combination at its free end are considered as shown in Fig. 10. In this example the composite box-beams made of variable stiffness layer-c and variable stiffness layer-d as defined in Table 7 are utilized. Here the variable stiffness layer-d being different from the other layers
Table 6 Maximum transverse displacement and axial rotation of eccentrically loaded variable stiffness box-beam with varying fibre orientations. 00
150
300
450
600
750
900
0 150 300 450 600 750 900
0,317 0,321 0,333 0,343 0,338 0,307 h(s) ? 1
0,325 0,341 0,368 0,392 0,404 0,401 0,372
0,352 0,381 0,419 0,451 0,466 0,471 0,467
0,387 0,422 0,462 0,488 0,498 0,501 0,498
0,412 0,448 0,482 0,501 0,506 0,506 0,504
0,424 0,459 0,489 0,502 0,506 0,505 0,504
h(s) ? 1 0,46 0,488 0,501 0,504 0,504 0,504
(h1); (h0)?
00
150
300
450
600
750
900
10,17 8,34 6,09 4,69 3,83 3,36 h(s) ? 1
8,22 5,96 4,31 3,46 3,05 3,05 3,58
5,99 4,28 3,28 2,84 2,76 3,09 4,02
4,71 3,46 2,84 2,68 2,85 3,47 4,78
3,98 3,08 2,77 2,85 3,29 4,32 6,10
3,57 3,06 3,07 3,48 4,35 6,02 8,36
h(s) ? 1 3,40 3,87 4,75 6,18 8,46 10,32
(h1); (h0)? 0
V (mm)
/ 10
3
(rad)
0
0 150 300 450 600 750 900
Fig. 9. Contour plots of maximum transverse displacement and axial rotation of eccentrically loaded variable stiffness box-beam.
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
Fig. 10. Variable stiffness box-beam loaded at its free end.
Table 7 Definitions of variable stiffness layer-c and -d, the corresponding fibre paths and the thickness changes. Layer
h0
h1
h2
h0
href
c d
10 10
45 45
10 45
0,2 mm 0,2 mm
45 45
defined previously is asymmetric with different h0 and h2 values. This asymmetric layer is placed around box-beam by providing the orientation angle continuity at the corners of the beam. The numerical results obtained for different loading conditions and lay-ups of layer-c and -d are presented in Table 8 and compared with the results of a finite element software (FEAS), Abaqus, where 4-node shell elements are used. As it is seen in Table 8 the results presented are very close to the ones obtained by FEAS. It should be noted that for a further refer-
ence the displacements were also obtained for the beams with different lay-ups of layer-d in their vertical and horizontal walls. As it can be observed from numerical results tabulated in Table 8, W0 and U are coupled under the effect of the load Fz and moment Mz for both lay-ups of layer-c and -d. It can also be said that applying only the load Fy for the beam with lay-up of layer-c causes almost no coupling between V and other displacements while the use of the layer-d in beams creates a strong coupling between U and V displacements due to asymmetric definition of this layer.
Table 8 Free end displacements and axial rotation of the variable stiffness composite box-beam for different loading conditions. Lay-Up
Load
Results
W0 (mm)
U (mm)
V (mm)
U (rad)
Top, Bottom, Left, Right ½þ þ c =þ c 5
Fy = 1 kN, Mz = 1 kNm, Fz = 100 kN
Present FEAS Present FEAS Present FEAS Present FEAS
2,861 2,838 0 0 2,869 2,844 0,009 0,007
0 0 0 0 0 0 0 0
10,8 10,9 10,8 10,9 0 0 0 0
0,1044 0,1041 0 0 0,0008 0,0007 0,1052 0,1047
Present FEAS Present FEAS Present FEAS Present FEAS
4,672 4,626 0 0 4,684 4,632 0,012 0,006
24,5 23,7 24,5 23,7 0 0 0 0
29,7 29,2 29,7 29,2 0 0 0 0
0,0837 0,0837 0 0 0,0013 0,0007 0,0851 0,0844
Fy = 1 kN Fz = 100 kN Mz = 1 kNm
Top, Bottom, ½þ þ d=þ d5 Left, Right ½þ d= d5
Fy = 1 kN, Mz = 1 kNm, Fz = 100 kN Fy = 1 kN Fz = 100 kN Mz = 1 kNm
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M. Gökhan Günay, T. Timarci / Composite Structures 182 (2017) 67–78
b66 ðsÞ ¼ B66 ðsÞ ½A22 ðsÞ B26 ðsÞ D26 ðsÞ B22 ðsÞfB26 ðsÞ2
6. Conclusion An analytical model was developed to investigate the flexuraltorsional behavior of single-cell closed cross-section thin -walled laminated composite beams with variable stiffness. Variable stiffness layers are acquired by using the curvilinear fibres. A procedure introduced for alignment and mapping of variable stiffness layers in a new scaled coordinate system facilitates to work with multiple variable stiffness layers. The symbolic notation presented makes it possible to show the variable stiffness layers in similar well-known lay-up notation used for classical constant stiffness layers. The numerical results of present model developed are initially obtained by use of a displacement based finite element method with 2-node beam elements for thin-walled having straight fibres and compared with the ones available in the literature After having seen the results agreed well with the ones of the previous constant stiffness case, new numerical results involving displacements and rotations for the same type of the beam with same conditions but with variable stiffness layers are also obtained and presented in order to see the effect of changing the orientation of fibre angle. The contour plots are presented to show that the same transverse displacement or axial rotation can be achieved for different fibre paths leading to a flexibility in the optimization process of such structures. Finally the composite box-beams made of two different fibre paths in their layers, clamped from one end and with four types of loading conditions are considered to investigate the resulting couplings between displacements. It is also observed that the numerical results obtained are in very good agreement with the results of the finite element analysis software. It can be said that using variable stiffness layers, that is, varying the paths of curvilinear fibres in structures provides more flexibility in their design to have a better mechanical performance in comparison with composite structures having classical straight fibres in their layers. Appendix A
þ A26 ðsÞ D26 ðsÞg þ A26 ðsÞ B26 ðsÞ D22 ðsÞ=D22 d11 ðsÞ ¼ D11 ðsÞ ½D12 ðsÞ2 A22 ðsÞ 2 D12 ðsÞ B12 ðsÞ B22 ðsÞ þ D22 ðsÞ B12 ðsÞ2 =D22 2 6 d16 ðsÞ ¼ D16 ðsÞ 4
2
þ A22 ðsÞ B12 ðsÞ =D22
ðA:1aÞ
þ B12 ðsÞ A26 ðsÞg þ A12 ðsÞ A26 ðsÞ D22 ðsÞ=D22
ðA:1bÞ
þ D22 ðsÞ B26 ðsÞ2 =D22
D22 ¼ A22 ðsÞ D22 ðsÞ B22 ðsÞ2
2
þ A22 ðsÞ B26 ðsÞ =D22
Z
l
k11 ¼ 0
Z
l
k12 ¼ 0
Z
l
k13 ¼ 0
Z
l
k14 ¼ 0
Z
l
k22 ¼
þ A12 ðsÞ D12 ðsÞg þ A22 ðsÞ B12 ðsÞ D12 ðsÞ=D22 6 a b16 ðsÞ ¼ B16 ðsÞ 4
A12 ðsÞ B26 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ B26 ðsÞ þA12 ðsÞ D26 ðsÞg
0
6 b b16 ðsÞ ¼ B16 ðsÞ 4
þA26 ðsÞ D12 ðsÞg
l
k33 ¼ 0 l
0
Z
l
0
ðA:1dÞ 3
Z f1 ¼
7 5=D22
3 7 5=D22
þA22 ðsÞ B26 ðsÞ D12 ðsÞ ðA:1fÞ
ðA:1jÞ ðA:2Þ
ðA:3aÞ
e12 S0L S00H
ðA:3bÞ
e13 S0L S00H
ðA:3cÞ
e15 S0L S0H e14 S0L S00H
ðA:3dÞ
e22 S00H S00H
ðA:3eÞ
e23 S00H S00H
ðA:3fÞ
e24 S00H S00H e25 S00H S0H
ðA:3gÞ
e33 S00H S00H
ðA:3hÞ
e34 S00H S00H e35 S00H S0H
ðA:3iÞ
e44 S00H S00H e45 ðS00H S0H þ S0H S00H Þ e55 S0H S0H
ðA:3jÞ
qz SL
ðA:4aÞ
qx SH
ðA:4bÞ
qy SH
ðA:4cÞ
mz SH
ðA:4dÞ
l
0
Z
e11 S0L S0L
l
0
Z f2 ¼
ðA:1eÞ A26 ðsÞ B12 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ B26 ðsÞ
l
k24 ¼
k44 ¼
þA22 ðsÞ B12 ðsÞ D26 ðsÞ 2
l
0
ðA:1cÞ
b11 ðsÞ ¼ B11 ðsÞ ½A12 ðsÞ B12 ðsÞ D22 ðsÞ B22 ðsÞfB12 ðsÞ 2
Z k23 ¼
k34 ¼
2
7 5=D22
Elements of element stiffness matrix and element force vector in Eqs. (33) and (34).
Z
a66 ðsÞ ¼ A66 ðsÞ ½A26 ðsÞ2 D22 ðsÞ 2 A26 ðsÞ B26 ðsÞ B22 ðsÞ
þD12 ðsÞ B26 ðsÞg
3
d66 ðsÞ ¼ D66 ðsÞ ½D26 ðsÞ2 A22 ðsÞ 2 D26 ðsÞ B26 ðsÞ B22 ðsÞ
Z
a16 ðsÞ ¼ A16 ðsÞ ½A22 ðsÞ B12 ðsÞ B26 ðsÞ B22 ðsÞfA12 ðsÞ B26 ðsÞ
A22 ðsÞ D12 ðsÞ D26 ðsÞ B22 ðsÞfB12 ðsÞ D26 ðsÞ
ðA:1iÞ
Z
a11 ðsÞ ¼ A11 ðsÞ ½A12 ðsÞ2 D22 ðsÞ 2 A12 ðsÞ B12 ðsÞ B22 ðsÞ
ðA:1hÞ
þB12 ðsÞ B26 ðsÞ D22 ðsÞ
0
The detailed expressions for the modified stiffness coefficients in Eq. (23):
ðA:1gÞ
l
f3 ¼ 0
Z f4 ¼ 0
l
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