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A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method
Composite Structures 96 (2013) 532–537
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method C.M.C. Roque a,⇑, D.S. Fidalgo a, A.J.M. Ferreira b, J.N. Reddy c a
INEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Departamento de Engenharia Mecânica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA b
a r t i c l e
i n f o
Article history: Available online 24 September 2012 Keywords: Composite Timoshenko beam Modified couple stress Nano beam
a b s t r a c t A modified couple stress theory and a meshless method are used to study the bending of simply supported laminated composite beams subjected to transverse loads. The Timoshenko beam kinematics are employed to model the beam, by a modified couple stress theory. The governing equations for the Timoshenko beam are solved numerically using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement and provides accurate results that are in excellent agreement with the analytical solutions. Moreover, the results show that the present model can capture the effects of the microstructure. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Experimental observations indicate that the mechanical behavior of micro and nano systems cannot be accurately simulated by using classical deformations theories. Alternatives to classical theories have been developed in order to account for scale effects observed at the micro and nano scales. Non classical models include strain gradient theory [1], nonlocal elasticity theory [2,3] and modified couple stress theory [4–9]. The modified couple stress theory has been mostly applied to beams and was previously used for the study of free vibrations analysis of micro plates [10–13] and static deflection of Kirchhoff [14] and Mindlin plates [7]. The modified couple stress theory has been used by many authors for the study of isotropic beams. Park and Gao [5] used a modified couple stress theory with an Euler Bernoulli formulation for the bending analysis of cantilever beams. Ma et al. [6] and Reddy [8] developed a modified Timoshenko beam theory and studied the bending and free vibration of simply supported beams using the Navier procedure. Asghari et al. [15] studied the nonlinear static and free vibrations of Timoshekso beams. Fu and Zhang studied the buckling of Timoshlenko beams [16]; Ma et al. [17] also developed a model for a Reddy–Levinson beam. Other applications include the analytical and numerical solution procedures for vibration of an embedded microbeam under action of a moving microparticle [18], resonant frequency and sensitivity of atomic force microscope (AFM) microcantilevers [19] and the stability of a microbeam conveying fluid [20]. Xia and ⇑ Corresponding author. E-mail address: [email protected] (C.M.C. Roque). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.09.011
Wang [21] and Xia et al. [22] used the modified Timoshenko beam model to analyze the free vibration of microstructures containing internal fluid flow and used a Euler Bernoulli modified beam model for nonlinear static bending, postbuckling and free vibration analyses, using the differential quadrature method. More recently the modified couple stress theory has been applied to the study of functionally graded beams using the modified Euler Bernoulli model for nonlinear analysis [23], the modified Timoshenko beam model [24], and a third order shear deformation beam theory for static and dynamic analysis of beams [25]. The differential quadrature method and the generalized differential quadrature method have been used to study the static, free vibration and buckling analysis of micro beams [26,27]. The analysis of composite micro beams using a modified couple stress theory has been proposed by Chen et al. [28,29] using the first and third order shear deformation theories. In the present paper, a modified couple stress theory is used to simulate the static bending of a micro composite laminated Timoshenko beam. The governing equations are solved, for the first time, using a meshless method based on collocation with radial basis functions. Also, analytical results obtained with the Navier method are presented for simply supported beams. The authors have previously used with success the radial basis function collocation meshless method to study the bending, buckling and free vibration of Timoshenko isotropic nanobeams, using a nonlocal elasticity theory [30]. 2. Modified couple stress theory The modified couple stress theory proposed by Yang et al. [4] has the advantage over the classical couple stress theory by involv-
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537
ing only a single scale parameter, l. A brief description of the modified couple stress theory is here presented. Further details can be found in [4,7,8]. The strain energy for an isotropic linearly elastic material in domain X is described as
X
e0k ð1Þ
where dV is the volume element, r is the symmetric part of the Cauchy stress tensor, e is the strain tensor, m is the deviatoric part of the couple stress tensor, v is the symmetric curvature tensor, k and l are Lamé constants and l is a material length scale. The kinematic relations are
1 2
1 2
e ¼ ½Du þ ðDuÞT ; v ¼ ½Dx þ ðDxÞT
ð2Þ
where u and x are the displacement and rotation vector, respectively, and x ¼ 12 curl u. Constitutive equations are described as 2
r ¼ k trðeÞI þ 2Ge; m ¼ 2‘ Gv
ð3Þ
where k and G are the Lamé constants
k¼
Em ; ð1 þ mÞð1 2mÞ
G¼
E 2ð1 þ mÞ
ð4Þ
3. Modified couple-stress theory equations for a composite laminated beam
uðx; zÞ ¼ u0 ðxÞ zhðxÞ
ð5Þ
v ðx; zÞ ¼ 0
ð6Þ
wðx; zÞ ¼ wðxÞ
ð7Þ
were h is the rotation about the y-axis. 3.1. Constitutive relations @x
@x
0
The curvatures @ yx0 0 and @ xy0 of the kth ply of a composite laminated beam layer with local coordinates ðx0 ; y0 ; zÞ, are expressed as independent components in terms of the micro-scale material constants, ‘2kb and ‘2km related to the fiber and matrix of the same ply, respectively. For the kth ply, the constitutive relations are given by [28],
m x0 y0 m y0 x0
"
¼
C k44 ‘2kb
C k55 ‘2km
#8 @ xx0 9 < 0 =
C k44 ‘2kb
C k55 ‘2km
: @ xy0 @ x0
@y
ð8Þ
;
where C k44 ¼ Gk13 and C k55 ¼ Gk23 . The stress moments are symmetric, i.e., m x0 y0 ¼ m y0 x0 and the curvatures,
@ xx0 @ y0
and
@ xy 0 @ x0
are asymmetric. The stress–strain relations
of kth layer are given by,
r0k ¼ C k e0k
ð9Þ
where
h
r0k ¼ rxk0 ryk0 sxk0 z syk0 z mxk0 y0 myk0 x0
iT
iT
ð10Þ
s is the shear stress and skxy is neglected and
8 @u0 9 8 9 > > > > @x0 0 e > > x > > > > @ m0 > > > > > > > > 0 > > > @y 0 > e > > > > y > > > > 0 > > > < c 0 = < @u þ @w0 > = @x @z xz ¼ ¼ @ m0 @w > @z þ @y0 > > cy0 z > > > > > > > > > > > > @ x x0 > > > > v x0 y0 > > > > > > > > > @y0 : ; > > > > > v y0 x0 : @xy0 > ;
ð11Þ
@x0
2
C k11
6 k 6 C 21 6 6 6 k C ¼6 6 6 6 4
3
C k12
7 7 7 7 7 7 7 7 k 7 2 ‘km C 55 5
C k22 C k44 C k55 ‘2kb C k44 ‘2kb C k44
‘2km C k55
mk Ek2 , mk 12 21
Ek
where C k11 ¼ 1mk1 mk , C k12 ¼ 1m12k 12
21
ð12Þ
Ek
C k22 ¼ 1mk2 mk , C k44 ¼ Gk13 and 12
21
C k55 ¼ Gk23 . Moreover, Ek1 and Ek2 are the elastic moduli; Gk13 and Gk23 are the shear moduli; mk12 and mk21 are the Poisson ratios;
mk21 Ek1 ¼ Ek2 mk12 ; ‘2kb and ‘2km are the material micro-structural constants related to the fiber and matrix [31]. The stress–strain relation of kth ply in the global coordinate system ðx; y; zÞ are given as follows:
rk ¼ Q k ek
Consider a n-ply composite laminated beam with domain X defined by x 2 ½0; L; y 2 ½0; b and z 2 ½h=2; þh=2, where displacements u; m and w are defined along the global x-, y- and z-axes, respectively. The x–y-plane ðz ¼ 0Þ is the undeformed mid-plane. The displacement field for the Timoshenko beam theory is defined as:
where
Z
1 ðr : e þ m : vÞdV 2 Z 1 2 kðtr eÞ2 þ l e : e þ l v : v dV ¼ X 2
U¼
h
e0k ¼ ex0 ey0 cx0 z cy0 z vx0 y0 vy0 x0
ð13Þ
where
h
rk ¼ rxk ryk sxzk skyz mxyk myxk h
e ¼ ex ey cxz cyz vxy vyx
iT
iT
ð14Þ
Q k ¼ T kT C k T k
ð15Þ k
The coordinate transformation matrix T is given by,
2 6 6 6 6 Tk ¼ 6 6 6 4
m2 n2 0 0 0 0
n2 m2 0 0 0 0
3 0 0 7 7 0 7 7 7 0 7 7 n2 5 m2
0 0 0 0 0 0 m n 0 n m 0 0 0 m2 0 0 n2
ð16Þ
where m ¼ cos /k , n ¼ sin /k and /k is the fiber angle related to the x-axis. As ey ¼ cyz ¼ 0, Q k becomes
2
Q k11
6 6 0 6 Qk ¼6 6 0 4 0
3
0
0
0
Q k44 0
0 bk ‘2 Q
0 bk ‘2 Q
0
bk ‘2k Q 44
bk ‘2k Q 55
k
44
k
7 7 7 7 7 55 5
ð17Þ
where
Q k11 ¼ m4 C k11 þ n4 C k22 þ 2m2 n2 C k12 þ 2C k66 Q k44 ¼ m2 C k44 þ n2 C k55 þ 2m2 n2 C k12 þ 2C k66 b k ¼ m4 ‘2 C k þ n4 ‘2 C k þ m2 n2 ‘2 C k þ ‘2 C k ‘2k Q kb 44 km 55 kb 44 km 55 44 b k ¼ n4 ‘2 C k þ m4 ‘2 C k þ m2 n2 ‘2 C k þ ‘2 C k ‘2k Q 55 kb 44 km 55 kb 44 km 55
ð18Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537 Table 2 Maximum rotation for composite Timoshenko beam subjected to sinusoidal distributed load, for ‘ ¼ 0; ‘ ¼ h=4, ‘ ¼ h=2, ‘ ¼ h; n-number of points, L=h ¼ 8. n
hmax ‘¼0
‘ ¼ h=4
‘ ¼ h=2
‘¼h
11 21 31 41 51 61 Analytical
0.0434 0.0440 0.0441 0.0441 0.0441 0.0441 0.0442
0.0431 0.0438 0.0439 0.0439 0.0439 0.0439 0.0439
0.0423 0.0430 0.0430 0.0430 0.0430 0.0430 0.0430
0.0394 0.0399 0.0399 0.0399 0.0397 0.0398 0.0400
In practice, ‘kb ‘km and so ‘km ¼ 0 is assumed. Then, Eq. (19) for cross-ply laminated beams can be simplified as:
Q k11 ¼ C k11 m4 þ C k22 n4 Fig. 1. Analytical and numerical solutions for beam deflection, with ‘ ¼ 0; h=4, h=2, h, n ¼ 30; L=h ¼ 8.
Q k44 ¼ C k44 m2 þ C k55 n2 b k ¼ ‘2 C k m4 ‘2 Q k
k
44
44
b k ¼ ‘2 C k n4 ‘2k Q k 44 55
ð20Þ
where ‘k ¼ ‘kb . For layers with the same material, ‘ ¼ ‘k . 3.2. Principle of virtual displacements The virtual work principle can be used to archive the equilibrium and boundary conditions. For a beam unitary width, i.e., b ¼ 1, the principle of virtual work for a composite laminated beam can be expressed by
dU ¼ dW
ð21Þ
where
dU ¼
Z
"
L
b
0
¼
n Z X
L
" n Z X b
0
Fig. 2. Analytical and numerical solutions beam rotation for ‘ ¼ 0; h=4, h=2, h, n ¼ 30; L=h ¼ 8.
dW ¼
Z
k¼1
zkþ1
xmax =h
n
11 21 31 41 51 61 Analytical
‘¼0
‘ ¼ h=4
‘ ¼ h=2
‘¼h
0.1852 0.1866 0.1867 0.1867 0.1867 0.1867 0.1867
0.1842 0.1856 0.1858 0.1858 0.1858 0.1858 0.1858
0.1815 0.1828 0.1829 0.1830 0.1830 0.1830 0.1830
0.1714 0.1725 0.1725 0.1725 0.1721 0.1722 0.1726
The mentioned formulations can be used to both isotropic beam and anisotropic beam. For the cross-ply laminates, /k ¼ 0 or p=2 which leads to mn ¼ 0 and Eq. (18) simplifies to
Q k11
4
¼m
C k11
4
Z
f du dV þ
þn
Q k44 ¼ m2 C k44 þ n2 C k55 b k ¼ m4 ‘2 C k þ n4 ‘2 C k ‘2k Q 44 kb 44 km 55 b k ¼ n4 ‘2 C k þ m4 ‘2 C k ‘2 Q k
55
kb
44
km
55
#
rkx dex þ skxz dcxz þ 2mkxy dvxy dz dx
ð22Þ
TT du dS
ð23Þ
@X
where f and T are the body force and boundary traction vectors, respectively, b is the width of the beam. Substituting for dU; dW, into the virtual work statement, noting that the virtual strains can be expressed in terms of the generalized displacements, integrating by parts to relieve from any derivatives of the generalized displacements and using the fundamental lemma of the calculus of variations, we obtain the Euler–Lagrange equations:
dN ¼0 dx 2 dQ 1 d Y þ þ fw ¼ 0 dx 2 dx2 dM 1 dY Q þ ¼0 dx 2 dx
ð24Þ
where
fN; M; Q; Y g ¼
n Z X k¼1
C k22
zk
X
Table 1 Maximum deflection for composite Timoshenko beam subjected to sinusoidal distributed load, for ‘ ¼ 0; ‘ ¼ h=4, ‘ ¼ h=2, ‘ ¼ h; n-number of points, L=h ¼ 8.
#
T
rk : de dz dx
zk
k¼1
Z
zkþ1
zkþ1
Zk
n
o
rkx ; zrkx ; skxz ; mkxy dz
ð25Þ
The traction boundary conditions at x ¼ 0 and x ¼ L can be obtained as
ð19Þ
N ¼ N;
Qþ
1 dY ¼ V; 2 dx
Y ¼ Y; 2
M
Y ¼M 2
ð26Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537
Fig. 3. Central deflection for various ratios L=h, with ‘ ¼ 0; h=2; h; n ¼ 30.
whereas the displacement boundary conditions are defined as
u0 ¼ u0 ;
w ¼ w;
Some of the most common RBFs are [32,33]: 1
dw dw ¼ ; dx dx
h¼h
ð27Þ
The equilibrium equations are expressed in terms of displacements as: 2
Q 11
d u0
2
d h
¼0 2 dx ! ! 2 3 4 d w dh ‘ 2 Q 44 d h d w Q 44 þ þ fw ¼ 0 2 3 4 dx 4 dx dx dx ! 2 2 2 3 d u0 d h dw ‘ 2 Q 44 d h d w ¼0 h J 11 I Q þ 11 44 2 2 2 3 dx 4 dx dx dx dx 2
dx
J 11
n X z3 z3k zzþ1 zk ; zþ1 Q kjj 3 k¼1 n 2 X zzþ1 z2k ¼ Q k11 2 k¼1
J 11
Q 44 ¼
ðj ¼ 1; 4Þ
n X fzzþ1 zk ;gQ k44
ð28Þ
ð29Þ ð30Þ
ð31Þ
ð32Þ
where
ð33Þ
The shape parameter c is a non-zero input parameter to be defined by the user and has a high influence in the solutions quality; d is the Euclidean distance between any two distinct grid points.
xj ¼ x xj
2
þ y þ yj
2
2
ð37Þ ð38Þ
LuðxÞ ¼ sðxÞ x 2 X Rn
ð39Þ
where L and B are differential operators in domain X and boundary @ X respectively. Points xj ; j ¼ 1; . . . ; N B and xj ; j ¼ N B þ 1; . . . ; N are used on the boundary and in the domain respectively. Consider~ of the solution uðxÞ ing an interpolant u
~ ðxÞ ¼ u
N X
aj g kx xj k
ð40Þ
and inserting L and B operators in Eq. (40) we obtain the following algebraic equations,
N > X > > > ~ L ðxÞ aj Lg kx xj k ¼ sðxi Þ i ¼ N B þ 1; . . . ; N u > :
ð41Þ
j¼1
where f ðxi Þ and sðxi Þ are the prescribed values on the boundary nodes and function values on the domain nodes, respectively. Solving the previous system in order to a, it is possible interpolate the solution by Eq. (40). 5. Navier analytical solutions
h i1=2 2 g j x xj ¼ dj x xj þ c2
2 dj x
jk
8 N X > > > ~ B ðxÞ aj Bg kx xj k ¼ f ðxi Þ i ¼ 1; . . . ; NB >u > < j¼1
j¼1
2 kxx
ð36Þ
j¼1
The collocation technique with radial basis functions (RBF) is applied for spatial approximations [32,33]. In this scheme it is assumed that any function, f may be written as an expression of N continuously differentiable basis functions, g:
aj g j x xj
Gaussians : g j ðxÞ ¼ ec
BuðxÞ ¼ f ðxÞ x 2 @ X Rn
4. Collocation with radial basis functions
N X
Inverse Multiquadrics : g j ðxÞ ¼ ðkx xj k þ c Þ
In this paper we will use the multiquadric radial basis function. Consider a boundary problem with domain X 2 Rn and with an elliptic differential equation given by,
k¼1
f ðxÞ ¼
ð35Þ 2 12
Thin Plate Splines : g j ðxÞ ¼ kx xj k2 log kx xj k
being
ðQ jj ; Ijj Þ ¼
Multiquadrics : g j ðxÞ ¼ ðkx xj k þ c2 Þ2
þ
ð34Þ
In order to assess the quality of solutions produced by the present numerical method, an analytical solution is computed using the Navier method, for simply supported beams of length b. The Navier method considers solutions of the form:
uðxÞ ¼
1 X U n ðxÞ n¼1
ð42Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537
Boundary conditions in (26) and (27) are satisfied by the following expansions of the displacements and applied transverse load:
fuðxÞ; /ðxÞg ¼
1 X fU n ; Un g cosðaxÞ
ð43Þ
n¼1
fwðxÞ; qðxÞg ¼
1 X fW n ; Q n g sinðaxÞ
ð44Þ
n¼1
with
a¼
np b
ð45Þ
Substituting Eqs. (43) and (44) in (28) we obtain the following algebraic set of equations:
KD ¼ F
ð46Þ
where D ¼ ðU n ; W n ; Un ÞT is the vector of generalized displacements, K is the stiffness matrix, and F is the vector of external applied forces. The analytical solution is then computed by solving the system of Eq. (46) for D and substituting U n ; W n ; Un in 43,44. 6. Discussion of numerical and analytical results A simply supported
h i 00 =900 =00 laminated cross-ply Timo-
shenko beam with width, b = 25 lm and constant thickness, h = 25 lm subjected to a sinusoidal load q0 ¼ 1:0 N=mm is considpffiffiffi ered. Shape parameter is of the form c ¼ 2L= n, where n is the number of points in a regular grid. Material properties are
E2 ¼ 6:98 GPa;
E1 ¼ 25E2 ;
G12 ¼ 0:5E2 ;
G22 ¼ 0:25E2 ;
m12 ¼ m22 ¼ 0:25 and EQ244 ¼ 0:5; bh
12I11 E2 bh3
¼ 25:72;
Q 44 E2 bh3
¼ 1=6. The shear correction factor
is chosen to be 1, since all solutions are compared with analytical solutions. In order to test the effect of the scale parameter, three different scale parameters are used ‘ ¼ h; ‘ ¼ h=2 and ‘ ¼ h=4 [28]. Figs. 1 and 2 show the numerical RBF and analytical solutions, for L=h ¼ 8. The present couples stress theory produces smaller deflections and rotation, when compared with classical solutions (‘ ¼ 0). This result is in line with results presented in [28] for composite beams. The numerical solutions are in excellent agreement with Navier analytical solutions. Table 1 and Table 2 present the maximum deformation and rotation, respectively, for different number of grid points. The numerical method is stable and converges rapidly to the analytical solution. In order to study the behavior of the method with thin and thick beams, Fig. 3 shows the central deflection for different ratios L=h (up to L=h ¼ 2000). Numerical and analytical Navier solutions are depicted for ‘ ¼ 0; h=2; h. The numerical method remains stable, even for very thin beams. 7. Conclusions A modified couple stress theory and a meshless method based on collocation with radial basis functions are used to study the static bending of composite laminated simply supported Timoshenko beams. The defection and rotation magnitudes of beams change according to the length scale parameter. Moreover, the numerical method provides accurate results, as can be seen by the agreement between analytical and numerical results.
Acknowledgement rio da Cie ^ncia, Tecnologia e do Ensino The support of Ministe Superior and Fundo Social Europeu (MCTES and FSE) under Programs POPH-QREN and Project PTDC/EME-PME/109116/2008 are gratefully acknowledged. References [1] Nix William D, Gao Huajian. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 1998;46(3):411–25. [2] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983;54(9):4703–10. [3] Eringen AC, Edelen DGB. On nonlocal elasticity. Int J Eng Sci 1972;10(3):233–48. [4] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002;39(10):2731–43. [5] Park SK, Gao X-L. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 2006;16(11):2355–9. [6] Ma HM, Gao X-L, Reddy JN. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 2008;56(12):3379–91. [7] Ma HM, Gao X-L, Reddy JN. A non-classical mindlin plate model based on a modified couple stress theory. Acta Mech 2011;220(1–4):217–35. [8] Reddy JN. Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 2011;59(11):2382–99. [9] Reddy JN, Arbind A. Bending relationships between the modified couple stressbased functionally graded Timoshenko beams and homogeneous Bernoulli– Euler beams. Ann Solid Struct Mech 2012:1–12. [10] Jomehzadeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E: Low-Dimens Syst Nanostruct 2011;43(4):877–83. [11] Ke L-L, Wang Y-S, Yang J, Kitipornchai S. Free vibration of size-dependent mindlin microplates based on the modified couple stress theory. J Sound Vib 2012;331(1):94–106. [12] Yin L, Qian Q, Wang L, Xia W. Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech Solida Sin 2010;23(5):386–93. [13] Chen W, Xu M, Li L. A model of composite laminated Reddy plate based on new modified couple stress theory. Compos Struct 2012;94(7):2143–56. [14] Tsiatas GC. A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 2009;46(13):2757–64. [15] Asghari M, Kahrobaiyan MH, Ahmadian MT. A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1749–61. [16] Fu Y, Zhang J. Modeling and analysis of microtubules based on a modified couple stress theory. Physica E: Low-Dimen Syst Nanostruct 2010;42(5):1741–5. [17] Ma HM, Gao X-L, Reddy JN. A nonclassical Reddy–Levinson beam model based on a modified couple stress theory. Int J Multisc Comput Eng 2010;8(2):167–80. [18] Simsek M. Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1721–32. [19] Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT. Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1985–94. [20] Ahangar S, Rezazadeh G, Shabani R, Ahmadi G, Toloei A. On the stability of a microbeam conveying fluid considering modified couple stress theory. Int J Mech Mater Des 2011;7(4):327–42. [21] Xia W, Wang L. Microfluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory. Microfluid Nanofluid 2010;9(4–5):955–62. [22] Xia W, Wang L, Yin L. Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 2010;48(12):2044–53. [23] Kahrobaiyan MH, Asghari M, Hoore M, Ahmadian MT. Nonlinear sizedependent forced vibrational behavior of microbeams based on a nonclassical continuum theory. JVC/J Vib Contr 2012;18(5):696–711. [24] Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT. The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 2011;32(3):1435–43. [25] Salamat-Talab M, Nateghi A, Torabi J. Static and dynamic analysis of thirdorder shear deformation fg micro beam based on modified couple stress theory. Int J Mech Sci 2012;57(1):63–73. [26] Ke L-L, Wang Y-S. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 2011;93(2):342–50. [27] Nateghi A, Salamat-talab M, Rezapour J, Daneshian B. Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory. Appl Math Modell 2012. [28] Wanji Chen, Chen Weiwei, Sze KY. A model of composite laminated Reddy beam based on a modified couple-stress theory. Compos Struct 2012(0).
C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537 [29] Chen W, Li L, Xu M. A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos Struct 2011;93(11):2723–32. [30] Roque CMC, Ferreira AJM, Reddy JN. Analysis of Timoshenko nanobeams with a nonlocal formulation meshless method. Int J Eng Sci 2011;49(9):976–84. [31] Reddy JN. Mechanics of laminated composite plates and shells. Theory and analysis. CRC Press; 2006.
537
[32] Kansa EJ. Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics. Part I: Surface approximations and partial derivative estimates. Comput Math Appl 1990;19(8–9):127–45. [33] Kansa EJ. Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – Part II: Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 1990;19(8/9):147–61.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method C.M.C. Roque a,⇑, D.S. Fidalgo a, A.J.M. Ferreira b, J.N. Reddy c a
INEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Departamento de Engenharia Mecânica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA b
a r t i c l e
i n f o
Article history: Available online 24 September 2012 Keywords: Composite Timoshenko beam Modified couple stress Nano beam
a b s t r a c t A modified couple stress theory and a meshless method are used to study the bending of simply supported laminated composite beams subjected to transverse loads. The Timoshenko beam kinematics are employed to model the beam, by a modified couple stress theory. The governing equations for the Timoshenko beam are solved numerically using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement and provides accurate results that are in excellent agreement with the analytical solutions. Moreover, the results show that the present model can capture the effects of the microstructure. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Experimental observations indicate that the mechanical behavior of micro and nano systems cannot be accurately simulated by using classical deformations theories. Alternatives to classical theories have been developed in order to account for scale effects observed at the micro and nano scales. Non classical models include strain gradient theory [1], nonlocal elasticity theory [2,3] and modified couple stress theory [4–9]. The modified couple stress theory has been mostly applied to beams and was previously used for the study of free vibrations analysis of micro plates [10–13] and static deflection of Kirchhoff [14] and Mindlin plates [7]. The modified couple stress theory has been used by many authors for the study of isotropic beams. Park and Gao [5] used a modified couple stress theory with an Euler Bernoulli formulation for the bending analysis of cantilever beams. Ma et al. [6] and Reddy [8] developed a modified Timoshenko beam theory and studied the bending and free vibration of simply supported beams using the Navier procedure. Asghari et al. [15] studied the nonlinear static and free vibrations of Timoshekso beams. Fu and Zhang studied the buckling of Timoshlenko beams [16]; Ma et al. [17] also developed a model for a Reddy–Levinson beam. Other applications include the analytical and numerical solution procedures for vibration of an embedded microbeam under action of a moving microparticle [18], resonant frequency and sensitivity of atomic force microscope (AFM) microcantilevers [19] and the stability of a microbeam conveying fluid [20]. Xia and ⇑ Corresponding author. E-mail address: [email protected] (C.M.C. Roque). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.09.011
Wang [21] and Xia et al. [22] used the modified Timoshenko beam model to analyze the free vibration of microstructures containing internal fluid flow and used a Euler Bernoulli modified beam model for nonlinear static bending, postbuckling and free vibration analyses, using the differential quadrature method. More recently the modified couple stress theory has been applied to the study of functionally graded beams using the modified Euler Bernoulli model for nonlinear analysis [23], the modified Timoshenko beam model [24], and a third order shear deformation beam theory for static and dynamic analysis of beams [25]. The differential quadrature method and the generalized differential quadrature method have been used to study the static, free vibration and buckling analysis of micro beams [26,27]. The analysis of composite micro beams using a modified couple stress theory has been proposed by Chen et al. [28,29] using the first and third order shear deformation theories. In the present paper, a modified couple stress theory is used to simulate the static bending of a micro composite laminated Timoshenko beam. The governing equations are solved, for the first time, using a meshless method based on collocation with radial basis functions. Also, analytical results obtained with the Navier method are presented for simply supported beams. The authors have previously used with success the radial basis function collocation meshless method to study the bending, buckling and free vibration of Timoshenko isotropic nanobeams, using a nonlocal elasticity theory [30]. 2. Modified couple stress theory The modified couple stress theory proposed by Yang et al. [4] has the advantage over the classical couple stress theory by involv-
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ing only a single scale parameter, l. A brief description of the modified couple stress theory is here presented. Further details can be found in [4,7,8]. The strain energy for an isotropic linearly elastic material in domain X is described as
X
e0k ð1Þ
where dV is the volume element, r is the symmetric part of the Cauchy stress tensor, e is the strain tensor, m is the deviatoric part of the couple stress tensor, v is the symmetric curvature tensor, k and l are Lamé constants and l is a material length scale. The kinematic relations are
1 2
1 2
e ¼ ½Du þ ðDuÞT ; v ¼ ½Dx þ ðDxÞT
ð2Þ
where u and x are the displacement and rotation vector, respectively, and x ¼ 12 curl u. Constitutive equations are described as 2
r ¼ k trðeÞI þ 2Ge; m ¼ 2‘ Gv
ð3Þ
where k and G are the Lamé constants
k¼
Em ; ð1 þ mÞð1 2mÞ
G¼
E 2ð1 þ mÞ
ð4Þ
3. Modified couple-stress theory equations for a composite laminated beam
uðx; zÞ ¼ u0 ðxÞ zhðxÞ
ð5Þ
v ðx; zÞ ¼ 0
ð6Þ
wðx; zÞ ¼ wðxÞ
ð7Þ
were h is the rotation about the y-axis. 3.1. Constitutive relations @x
@x
0
The curvatures @ yx0 0 and @ xy0 of the kth ply of a composite laminated beam layer with local coordinates ðx0 ; y0 ; zÞ, are expressed as independent components in terms of the micro-scale material constants, ‘2kb and ‘2km related to the fiber and matrix of the same ply, respectively. For the kth ply, the constitutive relations are given by [28],
m x0 y0 m y0 x0
"
¼
C k44 ‘2kb
C k55 ‘2km
#8 @ xx0 9 < 0 =
C k44 ‘2kb
C k55 ‘2km
: @ xy0 @ x0
@y
ð8Þ
;
where C k44 ¼ Gk13 and C k55 ¼ Gk23 . The stress moments are symmetric, i.e., m x0 y0 ¼ m y0 x0 and the curvatures,
@ xx0 @ y0
and
@ xy 0 @ x0
are asymmetric. The stress–strain relations
of kth layer are given by,
r0k ¼ C k e0k
ð9Þ
where
h
r0k ¼ rxk0 ryk0 sxk0 z syk0 z mxk0 y0 myk0 x0
iT
iT
ð10Þ
s is the shear stress and skxy is neglected and
8 @u0 9 8 9 > > > > @x0 0 e > > x > > > > @ m0 > > > > > > > > 0 > > > @y 0 > e > > > > y > > > > 0 > > > < c 0 = < @u þ @w0 > = @x @z xz ¼ ¼ @ m0 @w > @z þ @y0 > > cy0 z > > > > > > > > > > > > @ x x0 > > > > v x0 y0 > > > > > > > > > @y0 : ; > > > > > v y0 x0 : @xy0 > ;
ð11Þ
@x0
2
C k11
6 k 6 C 21 6 6 6 k C ¼6 6 6 6 4
3
C k12
7 7 7 7 7 7 7 7 k 7 2 ‘km C 55 5
C k22 C k44 C k55 ‘2kb C k44 ‘2kb C k44
‘2km C k55
mk Ek2 , mk 12 21
Ek
where C k11 ¼ 1mk1 mk , C k12 ¼ 1m12k 12
21
ð12Þ
Ek
C k22 ¼ 1mk2 mk , C k44 ¼ Gk13 and 12
21
C k55 ¼ Gk23 . Moreover, Ek1 and Ek2 are the elastic moduli; Gk13 and Gk23 are the shear moduli; mk12 and mk21 are the Poisson ratios;
mk21 Ek1 ¼ Ek2 mk12 ; ‘2kb and ‘2km are the material micro-structural constants related to the fiber and matrix [31]. The stress–strain relation of kth ply in the global coordinate system ðx; y; zÞ are given as follows:
rk ¼ Q k ek
Consider a n-ply composite laminated beam with domain X defined by x 2 ½0; L; y 2 ½0; b and z 2 ½h=2; þh=2, where displacements u; m and w are defined along the global x-, y- and z-axes, respectively. The x–y-plane ðz ¼ 0Þ is the undeformed mid-plane. The displacement field for the Timoshenko beam theory is defined as:
where
Z
1 ðr : e þ m : vÞdV 2 Z 1 2 kðtr eÞ2 þ l e : e þ l v : v dV ¼ X 2
U¼
h
e0k ¼ ex0 ey0 cx0 z cy0 z vx0 y0 vy0 x0
ð13Þ
where
h
rk ¼ rxk ryk sxzk skyz mxyk myxk h
e ¼ ex ey cxz cyz vxy vyx
iT
iT
ð14Þ
Q k ¼ T kT C k T k
ð15Þ k
The coordinate transformation matrix T is given by,
2 6 6 6 6 Tk ¼ 6 6 6 4
m2 n2 0 0 0 0
n2 m2 0 0 0 0
3 0 0 7 7 0 7 7 7 0 7 7 n2 5 m2
0 0 0 0 0 0 m n 0 n m 0 0 0 m2 0 0 n2
ð16Þ
where m ¼ cos /k , n ¼ sin /k and /k is the fiber angle related to the x-axis. As ey ¼ cyz ¼ 0, Q k becomes
2
Q k11
6 6 0 6 Qk ¼6 6 0 4 0
3
0
0
0
Q k44 0
0 bk ‘2 Q
0 bk ‘2 Q
0
bk ‘2k Q 44
bk ‘2k Q 55
k
44
k
7 7 7 7 7 55 5
ð17Þ
where
Q k11 ¼ m4 C k11 þ n4 C k22 þ 2m2 n2 C k12 þ 2C k66 Q k44 ¼ m2 C k44 þ n2 C k55 þ 2m2 n2 C k12 þ 2C k66 b k ¼ m4 ‘2 C k þ n4 ‘2 C k þ m2 n2 ‘2 C k þ ‘2 C k ‘2k Q kb 44 km 55 kb 44 km 55 44 b k ¼ n4 ‘2 C k þ m4 ‘2 C k þ m2 n2 ‘2 C k þ ‘2 C k ‘2k Q 55 kb 44 km 55 kb 44 km 55
ð18Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537 Table 2 Maximum rotation for composite Timoshenko beam subjected to sinusoidal distributed load, for ‘ ¼ 0; ‘ ¼ h=4, ‘ ¼ h=2, ‘ ¼ h; n-number of points, L=h ¼ 8. n
hmax ‘¼0
‘ ¼ h=4
‘ ¼ h=2
‘¼h
11 21 31 41 51 61 Analytical
0.0434 0.0440 0.0441 0.0441 0.0441 0.0441 0.0442
0.0431 0.0438 0.0439 0.0439 0.0439 0.0439 0.0439
0.0423 0.0430 0.0430 0.0430 0.0430 0.0430 0.0430
0.0394 0.0399 0.0399 0.0399 0.0397 0.0398 0.0400
In practice, ‘kb ‘km and so ‘km ¼ 0 is assumed. Then, Eq. (19) for cross-ply laminated beams can be simplified as:
Q k11 ¼ C k11 m4 þ C k22 n4 Fig. 1. Analytical and numerical solutions for beam deflection, with ‘ ¼ 0; h=4, h=2, h, n ¼ 30; L=h ¼ 8.
Q k44 ¼ C k44 m2 þ C k55 n2 b k ¼ ‘2 C k m4 ‘2 Q k
k
44
44
b k ¼ ‘2 C k n4 ‘2k Q k 44 55
ð20Þ
where ‘k ¼ ‘kb . For layers with the same material, ‘ ¼ ‘k . 3.2. Principle of virtual displacements The virtual work principle can be used to archive the equilibrium and boundary conditions. For a beam unitary width, i.e., b ¼ 1, the principle of virtual work for a composite laminated beam can be expressed by
dU ¼ dW
ð21Þ
where
dU ¼
Z
"
L
b
0
¼
n Z X
L
" n Z X b
0
Fig. 2. Analytical and numerical solutions beam rotation for ‘ ¼ 0; h=4, h=2, h, n ¼ 30; L=h ¼ 8.
dW ¼
Z
k¼1
zkþ1
xmax =h
n
11 21 31 41 51 61 Analytical
‘¼0
‘ ¼ h=4
‘ ¼ h=2
‘¼h
0.1852 0.1866 0.1867 0.1867 0.1867 0.1867 0.1867
0.1842 0.1856 0.1858 0.1858 0.1858 0.1858 0.1858
0.1815 0.1828 0.1829 0.1830 0.1830 0.1830 0.1830
0.1714 0.1725 0.1725 0.1725 0.1721 0.1722 0.1726
The mentioned formulations can be used to both isotropic beam and anisotropic beam. For the cross-ply laminates, /k ¼ 0 or p=2 which leads to mn ¼ 0 and Eq. (18) simplifies to
Q k11
4
¼m
C k11
4
Z
f du dV þ
þn
Q k44 ¼ m2 C k44 þ n2 C k55 b k ¼ m4 ‘2 C k þ n4 ‘2 C k ‘2k Q 44 kb 44 km 55 b k ¼ n4 ‘2 C k þ m4 ‘2 C k ‘2 Q k
55
kb
44
km
55
#
rkx dex þ skxz dcxz þ 2mkxy dvxy dz dx
ð22Þ
TT du dS
ð23Þ
@X
where f and T are the body force and boundary traction vectors, respectively, b is the width of the beam. Substituting for dU; dW, into the virtual work statement, noting that the virtual strains can be expressed in terms of the generalized displacements, integrating by parts to relieve from any derivatives of the generalized displacements and using the fundamental lemma of the calculus of variations, we obtain the Euler–Lagrange equations:
dN ¼0 dx 2 dQ 1 d Y þ þ fw ¼ 0 dx 2 dx2 dM 1 dY Q þ ¼0 dx 2 dx
ð24Þ
where
fN; M; Q; Y g ¼
n Z X k¼1
C k22
zk
X
Table 1 Maximum deflection for composite Timoshenko beam subjected to sinusoidal distributed load, for ‘ ¼ 0; ‘ ¼ h=4, ‘ ¼ h=2, ‘ ¼ h; n-number of points, L=h ¼ 8.
#
T
rk : de dz dx
zk
k¼1
Z
zkþ1
zkþ1
Zk
n
o
rkx ; zrkx ; skxz ; mkxy dz
ð25Þ
The traction boundary conditions at x ¼ 0 and x ¼ L can be obtained as
ð19Þ
N ¼ N;
Qþ
1 dY ¼ V; 2 dx
Y ¼ Y; 2
M
Y ¼M 2
ð26Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537
Fig. 3. Central deflection for various ratios L=h, with ‘ ¼ 0; h=2; h; n ¼ 30.
whereas the displacement boundary conditions are defined as
u0 ¼ u0 ;
w ¼ w;
Some of the most common RBFs are [32,33]: 1
dw dw ¼ ; dx dx
h¼h
ð27Þ
The equilibrium equations are expressed in terms of displacements as: 2
Q 11
d u0
2
d h
¼0 2 dx ! ! 2 3 4 d w dh ‘ 2 Q 44 d h d w Q 44 þ þ fw ¼ 0 2 3 4 dx 4 dx dx dx ! 2 2 2 3 d u0 d h dw ‘ 2 Q 44 d h d w ¼0 h J 11 I Q þ 11 44 2 2 2 3 dx 4 dx dx dx dx 2
dx
J 11
n X z3 z3k zzþ1 zk ; zþ1 Q kjj 3 k¼1 n 2 X zzþ1 z2k ¼ Q k11 2 k¼1
J 11
Q 44 ¼
ðj ¼ 1; 4Þ
n X fzzþ1 zk ;gQ k44
ð28Þ
ð29Þ ð30Þ
ð31Þ
ð32Þ
where
ð33Þ
The shape parameter c is a non-zero input parameter to be defined by the user and has a high influence in the solutions quality; d is the Euclidean distance between any two distinct grid points.
xj ¼ x xj
2
þ y þ yj
2
2
ð37Þ ð38Þ
LuðxÞ ¼ sðxÞ x 2 X Rn
ð39Þ
where L and B are differential operators in domain X and boundary @ X respectively. Points xj ; j ¼ 1; . . . ; N B and xj ; j ¼ N B þ 1; . . . ; N are used on the boundary and in the domain respectively. Consider~ of the solution uðxÞ ing an interpolant u
~ ðxÞ ¼ u
N X
aj g kx xj k
ð40Þ
and inserting L and B operators in Eq. (40) we obtain the following algebraic equations,
N > X > > > ~ L ðxÞ aj Lg kx xj k ¼ sðxi Þ i ¼ N B þ 1; . . . ; N u > :
ð41Þ
j¼1
where f ðxi Þ and sðxi Þ are the prescribed values on the boundary nodes and function values on the domain nodes, respectively. Solving the previous system in order to a, it is possible interpolate the solution by Eq. (40). 5. Navier analytical solutions
h i1=2 2 g j x xj ¼ dj x xj þ c2
2 dj x
jk
8 N X > > > ~ B ðxÞ aj Bg kx xj k ¼ f ðxi Þ i ¼ 1; . . . ; NB >u > < j¼1
j¼1
2 kxx
ð36Þ
j¼1
The collocation technique with radial basis functions (RBF) is applied for spatial approximations [32,33]. In this scheme it is assumed that any function, f may be written as an expression of N continuously differentiable basis functions, g:
aj g j x xj
Gaussians : g j ðxÞ ¼ ec
BuðxÞ ¼ f ðxÞ x 2 @ X Rn
4. Collocation with radial basis functions
N X
Inverse Multiquadrics : g j ðxÞ ¼ ðkx xj k þ c Þ
In this paper we will use the multiquadric radial basis function. Consider a boundary problem with domain X 2 Rn and with an elliptic differential equation given by,
k¼1
f ðxÞ ¼
ð35Þ 2 12
Thin Plate Splines : g j ðxÞ ¼ kx xj k2 log kx xj k
being
ðQ jj ; Ijj Þ ¼
Multiquadrics : g j ðxÞ ¼ ðkx xj k þ c2 Þ2
þ
ð34Þ
In order to assess the quality of solutions produced by the present numerical method, an analytical solution is computed using the Navier method, for simply supported beams of length b. The Navier method considers solutions of the form:
uðxÞ ¼
1 X U n ðxÞ n¼1
ð42Þ
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C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537
Boundary conditions in (26) and (27) are satisfied by the following expansions of the displacements and applied transverse load:
fuðxÞ; /ðxÞg ¼
1 X fU n ; Un g cosðaxÞ
ð43Þ
n¼1
fwðxÞ; qðxÞg ¼
1 X fW n ; Q n g sinðaxÞ
ð44Þ
n¼1
with
a¼
np b
ð45Þ
Substituting Eqs. (43) and (44) in (28) we obtain the following algebraic set of equations:
KD ¼ F
ð46Þ
where D ¼ ðU n ; W n ; Un ÞT is the vector of generalized displacements, K is the stiffness matrix, and F is the vector of external applied forces. The analytical solution is then computed by solving the system of Eq. (46) for D and substituting U n ; W n ; Un in 43,44. 6. Discussion of numerical and analytical results A simply supported
h i 00 =900 =00 laminated cross-ply Timo-
shenko beam with width, b = 25 lm and constant thickness, h = 25 lm subjected to a sinusoidal load q0 ¼ 1:0 N=mm is considpffiffiffi ered. Shape parameter is of the form c ¼ 2L= n, where n is the number of points in a regular grid. Material properties are
E2 ¼ 6:98 GPa;
E1 ¼ 25E2 ;
G12 ¼ 0:5E2 ;
G22 ¼ 0:25E2 ;
m12 ¼ m22 ¼ 0:25 and EQ244 ¼ 0:5; bh
12I11 E2 bh3
¼ 25:72;
Q 44 E2 bh3
¼ 1=6. The shear correction factor
is chosen to be 1, since all solutions are compared with analytical solutions. In order to test the effect of the scale parameter, three different scale parameters are used ‘ ¼ h; ‘ ¼ h=2 and ‘ ¼ h=4 [28]. Figs. 1 and 2 show the numerical RBF and analytical solutions, for L=h ¼ 8. The present couples stress theory produces smaller deflections and rotation, when compared with classical solutions (‘ ¼ 0). This result is in line with results presented in [28] for composite beams. The numerical solutions are in excellent agreement with Navier analytical solutions. Table 1 and Table 2 present the maximum deformation and rotation, respectively, for different number of grid points. The numerical method is stable and converges rapidly to the analytical solution. In order to study the behavior of the method with thin and thick beams, Fig. 3 shows the central deflection for different ratios L=h (up to L=h ¼ 2000). Numerical and analytical Navier solutions are depicted for ‘ ¼ 0; h=2; h. The numerical method remains stable, even for very thin beams. 7. Conclusions A modified couple stress theory and a meshless method based on collocation with radial basis functions are used to study the static bending of composite laminated simply supported Timoshenko beams. The defection and rotation magnitudes of beams change according to the length scale parameter. Moreover, the numerical method provides accurate results, as can be seen by the agreement between analytical and numerical results.
Acknowledgement rio da Cie ^ncia, Tecnologia e do Ensino The support of Ministe Superior and Fundo Social Europeu (MCTES and FSE) under Programs POPH-QREN and Project PTDC/EME-PME/109116/2008 are gratefully acknowledged. References [1] Nix William D, Gao Huajian. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 1998;46(3):411–25. [2] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983;54(9):4703–10. [3] Eringen AC, Edelen DGB. On nonlocal elasticity. Int J Eng Sci 1972;10(3):233–48. [4] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002;39(10):2731–43. [5] Park SK, Gao X-L. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 2006;16(11):2355–9. [6] Ma HM, Gao X-L, Reddy JN. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 2008;56(12):3379–91. [7] Ma HM, Gao X-L, Reddy JN. A non-classical mindlin plate model based on a modified couple stress theory. Acta Mech 2011;220(1–4):217–35. [8] Reddy JN. Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 2011;59(11):2382–99. [9] Reddy JN, Arbind A. Bending relationships between the modified couple stressbased functionally graded Timoshenko beams and homogeneous Bernoulli– Euler beams. Ann Solid Struct Mech 2012:1–12. [10] Jomehzadeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E: Low-Dimens Syst Nanostruct 2011;43(4):877–83. [11] Ke L-L, Wang Y-S, Yang J, Kitipornchai S. Free vibration of size-dependent mindlin microplates based on the modified couple stress theory. J Sound Vib 2012;331(1):94–106. [12] Yin L, Qian Q, Wang L, Xia W. Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech Solida Sin 2010;23(5):386–93. [13] Chen W, Xu M, Li L. A model of composite laminated Reddy plate based on new modified couple stress theory. Compos Struct 2012;94(7):2143–56. [14] Tsiatas GC. A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 2009;46(13):2757–64. [15] Asghari M, Kahrobaiyan MH, Ahmadian MT. A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1749–61. [16] Fu Y, Zhang J. Modeling and analysis of microtubules based on a modified couple stress theory. Physica E: Low-Dimen Syst Nanostruct 2010;42(5):1741–5. [17] Ma HM, Gao X-L, Reddy JN. A nonclassical Reddy–Levinson beam model based on a modified couple stress theory. Int J Multisc Comput Eng 2010;8(2):167–80. [18] Simsek M. Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1721–32. [19] Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT. Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1985–94. [20] Ahangar S, Rezazadeh G, Shabani R, Ahmadi G, Toloei A. On the stability of a microbeam conveying fluid considering modified couple stress theory. Int J Mech Mater Des 2011;7(4):327–42. [21] Xia W, Wang L. Microfluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory. Microfluid Nanofluid 2010;9(4–5):955–62. [22] Xia W, Wang L, Yin L. Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 2010;48(12):2044–53. [23] Kahrobaiyan MH, Asghari M, Hoore M, Ahmadian MT. Nonlinear sizedependent forced vibrational behavior of microbeams based on a nonclassical continuum theory. JVC/J Vib Contr 2012;18(5):696–711. [24] Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT. The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 2011;32(3):1435–43. [25] Salamat-Talab M, Nateghi A, Torabi J. Static and dynamic analysis of thirdorder shear deformation fg micro beam based on modified couple stress theory. Int J Mech Sci 2012;57(1):63–73. [26] Ke L-L, Wang Y-S. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 2011;93(2):342–50. [27] Nateghi A, Salamat-talab M, Rezapour J, Daneshian B. Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory. Appl Math Modell 2012. [28] Wanji Chen, Chen Weiwei, Sze KY. A model of composite laminated Reddy beam based on a modified couple-stress theory. Compos Struct 2012(0).
C.M.C. Roque et al. / Composite Structures 96 (2013) 532–537 [29] Chen W, Li L, Xu M. A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos Struct 2011;93(11):2723–32. [30] Roque CMC, Ferreira AJM, Reddy JN. Analysis of Timoshenko nanobeams with a nonlocal formulation meshless method. Int J Eng Sci 2011;49(9):976–84. [31] Reddy JN. Mechanics of laminated composite plates and shells. Theory and analysis. CRC Press; 2006.
537
[32] Kansa EJ. Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics. Part I: Surface approximations and partial derivative estimates. Comput Math Appl 1990;19(8–9):127–45. [33] Kansa EJ. Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – Part II: Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 1990;19(8/9):147–61.