Voc, Huu-Tai Thaid a Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam b Faculty o
Trang 1Contents lists available atScienceDirect
Composite Structures journal homepage:www.elsevier.com/locate/compstruct
Trung-Kien Nguyena,⁎, Ba-Duy Nguyena,b, Thuc P Voc, Huu-Tai Thaid
a Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam
b Faculty of Civil Engineering, Thu Dau Mot University, 6 Tran Van On Street, Phu Hoa District, Thu Dau Mot City, Binh Duong Province, Viet Nam
c School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, VIC 3086, Australia
d Department of Infrastructure Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
A R T I C L E I N F O
Keywords:
Shear deformation beam theory
Laminated composite beam
Vibration
Bending
Buckling
A B S T R A C T
Based on fundamental equations of the elasticity theory, a novel unified beam model is developed for laminated composite beams In this model, the displacementfield is selected in a unified form which can be recovered to that of existing shear deformation beam theories available in the literature Based on Lagrange’s equations, the governing equations of the present theory are derived They are then solved for deflections, stresses, natural frequencies and critical buckling loads of composite beams under different boundary conditions and lay-ups by using the Ritz approach with novel hybrid trigonometric functions Various examples are also presented to verify the accuracy and generalization of the present theory, as well as investigate the influences of fibre angle on the behaviour of composite beams under different boundary conditions and lay-ups
1 Introduction
Laminated composite materials are commonly used in spacecraft,
aircraft, mechanical engineering, construction and other different
en-gineeringfields due to their excellent mechanical properties including
high strength, high stiffness and lightweight The widespread
applica-tions of these structures led to the development of different
computa-tional models to predict their behaviours
A general review of theories to analyse the laminated composite
beams can be found in the previous works[1–5] Generally, their
be-haviours can be captured based on either 2D beam theories or 3D
theory of elasticity Thefirst approach is more popular due to its
sim-plicity, whilst the second one is complicated to implement although it
can analyse exactly response Based on thefirst method, a huge number
of shear deformation models have been developed for composite beams
The simplest one is the first-order shear deformation beam theory
(FSBT) which requires a shear correction factor to compensate for the
inadequate distribution of shear stress The higher-order shear
de-formation beam theory (HSBT) with higher-order variations of axial
displacement gives slightly improved predictions compared with FSBT
However, it involves more number of unknowns and thus is more
complicated than FSBT Another shear deformation beam theory
de-veloped based on thefirst approach is quasi-3D theory where both axial
and transverse displacements are approximated as high-order
varia-tions through the beam thickness Therefore, quasi-3D theory can
predict the behaviour of composite beams more accurately than HSBT However, quasi-3D theory is more complicated than HSBT because it involves more numbers of unknowns Both HSBT and quasi-3D theory
do not require the shear correction factor However, their accuracy strictly depends on a choice of shear functions The development of shear functions for these theories is therefore an interesting topic that has attracted many researches with different approaches Various types
of shear functions have been developed for composite plates such as polynomial ([6–10]), trigonometric ([11–17]), exponential ([18]), hy-perbolic ([19,20]), and hybrid ([21,22]) For laminated composite beams, only some representative references are herein cited For ex-ample, Khdeir and Reddy [23,24] derived closed-form solutions of Reddy’s theory for critical buckling loads and natural frequencies of cross-ply composite beams Chandrashekhara and Bangera [25], Shi and Lam [26,27], Murthy et al.[28] and Manur and Kant[29] in-vestigated vibration behaviours of composite beams viafinite element method (FEM) Karama et al.[30,31]examined static, buckling and free vibration responses of composite beams based on trigonometric theory Aydogdu [32–34] used polynomial, hyperbolic and exponential the-ories and the Ritz method to explore the behaviour of composite beams Shao et al [35] presented various HSBTs for the free vibration of composite beams Vo and Thai[36–39]presented FEM solutions of both HSBT and quasi-3D theory for the structural analysis of composite beams Mantari and Canales [40,41] developed both Ritz and FEM solutions for composite beams by using the quasi-3D theories with a
https://doi.org/10.1016/j.compstruct.2020.111943
Received 9 October 2019; Received in revised form 9 December 2019; Accepted 16 January 2020
⁎Corresponding author
E-mail address:kiennt@hcmute.edu.vn(T.-K Nguyen)
Available online 25 January 2020
0263-8223/ © 2020 Elsevier Ltd All rights reserved
T
Trang 2third-order, polynomial and hybrid polynomial-trigonometric theories.
Matsunaga [42] considered free vibration responses of composite
beams by means of quasi-3D theory and Navier procedure By using
Carrera Unified Formulation (CUF) [43], Carrera et al [44–47] and
Arruda et al [48]analysed laminated composite beams Vidal et al
[49] recently developed higher-order beam elements to model
com-posite beams with generic cross-section
This paper aims to propose a novel unified HSBT which can be
re-covered to existing HSBT by changing the shear functions The Ritz
solution method with novel hybrid shape functions is employed to
de-velop approximate solutions for deflections, stresses, critical buckling
loads and natural frequencies of laminated composite beams under
various boundary conditions and lay-ups In order to validate the
ac-curacy of the proposed theory, several numerical examples in static,
vibration and buckling are considered In addition, the effects of fibre
orientation on the behaviour of laminated composite beams are also
examined
2 Theoretical formulation
2.1 A general framework of higher-order displacementfield
For the simplicity purpose, it is supposed that the effects of the
transverse normal strain is neglected and all formulations in this section
are performed under the assumption of isotropic materials The stress
-strain relationships of the beam are therefore given by:
∊ =
E σ
=
γ
G σ
in which E and G are Young’s modulus and shear modulus, respectively
Moreover, the linear elastic relationships of strains and displacements
are expressed by:
in which u and w are respectively axial and transverse deflections
Substituting Eq (2) into Eq (1) leads to:
=
u
E σ
x x
G σ
z x xz
The transverse shear stress σ xz( , )x z can be expressed in terms of the
transverse shear force Q x x( )as follows:
=
whereg z( )is a higher-order shear function that satisfies the
traction-free conditions at the bottom and top surfaces of the beam, i.e
g z( h/2) 0with h being the beam thickness Substituting Eq.(4)
into Eq (3b) and taking into account the hypothesis ofw x z( , )=w x0( ),
the axial displacement of the beam can be expressed by:
⎛
⎝
⎞
⎠
u x z u x zw f z Q x
G
, 0( ) 0,x ( ) x( )
(5) where ′f z( )=g z( )
Therefore, a general formulation of the displacement field of the
proposed theory is obtained as follows:
⎛
⎝
⎞
⎠
u x z u x zw f z Q x
G
, 0( ) 0,x ( ) x( )
(6a)
=
It can be noticed that whenf z( )= (z− )
h z h
3 2 4 3
3
2 , Eq (6) yields to the zeroth-order shear deformation theory developed by Shimpi[50]for plates Moreover, if the shear force is expressed in the form:
=
whereθ0is the beam rotation, the displacementfield in Eq (6) becomes
a common form of the HSBT, which has been widely used by many authors due to its simplicity:
u x z( , ) u x0( ) w0,x z f z θ x( ) 0( ) (8a)
=
Furthermore, if the transverse shear force is written in terms of the rotation and the derivative of transverse deflection as follows ([7]):
Q 5Gh θ w
x 0 0,x
(9)
Eq (6) becomes a general formulation of the HSBT as follows:
⎛
⎝
⎞
⎠
⎝
− ⎞
⎠ +
⎝
− ⎞
⎠ +
u x z u x hf z w hf θ x
6
5
6 ( )
x
x
(10a)
=
whereΦ( )z =5h f z( )
6 The displacementfield in Eq (10) can be considered as an unified HSBT that many theories can be obtained For example, the HSBT proposed by Reissner[7], Shi[26]and Reddy[9]can be recovered with
f z( ) z
h z h
3 2 4 3
3
2 and f z( )= (z− )
h z h
6 5 4 3
3
2 , respectively
2.2 Strain and stressfields The non-zero strains associated to the displacements in Eq (10) are therefore given by:
∊x( , )x z =u0,x+(Φ−z w) 0,xx+Φθ0,x (11a)
The non-zero stresses at thekth-layer of the beam (Fig 1) associated
to the strains in Eq (11) are given by ([5]):
σ x z x( , ) Q11 x Q11[u0,x (Φ z w) 0,xx Φθ0,x] (12a)
σ xz( , )x z Q γ55 xz Q55Φ (θ0 w0,x) (12b) where
Q ,Q Q
11 11 13
2
33
55 55
(13a)
= ′ + ′ ′ − ′ ′ ′ + ′ ′
′ − ′ ′
2
11 11 16
2
22 12 16 26 122 66
Fig 1 Geometric dimensions of a typical laminated composite beam
Trang 3= ′ + ′ ′ ′ + ′ ′ ′ − ′ ′ ′ − ′ ′ ′
′ − ′ ′
13 13 16 22 36 12 23 66 16 23 26 12 26 36
= ′ + ′ ′ − ′ ′ ′ + ′ ′
′ − ′ ′
2
33 33 36
2
22 23 26 36 232 66
= ′ − ′
′
C
55 55 45
2
where ′C ijare the elastic stiffness coefficients at the kth-layer of the beam
(see[51]for more details)
2.3 Governing equations of motion
The strain energy of the proposed beam model can be written as:
∫
∫
+
σ σ γ dV
H θ
V x x xz xz
L
x x xx xx s x x s xx x
s
x
s
1
2
1
2 0 0,
2
0, 0, 0,2 0, 0, 0, 0,
0,2
0 0,2 0 0,
U
(14)
where (A A B B D D, s, , s, , sandH s) are the stiffness coefficients defined
as:
∫
−
A B D B D H
{ , , , , , }
{1, Φ , (Φ ) , Φ, (Φ )Φ, Φ }
s s s
h
h
/2
/2
∫
−
A s Φ Q bdz
h
h
/2
/2 2
The external work done by the transverse load q and axial
com-pressionN0is obtained as follows:
= − qw dx−1 N w dx
2
x
2
V
(17) The kinetic energy of the proposed beam model is calculated by:
∫
∫
+
ρ z u w dV
I u I u w I w J θ u J θ w K θ
I w dx
( )( ̇ ̇ )
̇ ]
V
L
1
2
2 2
1
2 0 0 0 1 0 0, 2 0,
2
1 0 0 2 0 0, 2 0
2
0 0
K
(18)
in which dot-superscript indicates time derivativet ρ; is the mass
den-sity; and the moment of inertia terms I I I J J K0, ,1 2, 1, 2, 2are defined as:
∫
−
{ , , , , , } {1, Φ , (Φ ) , Φ, (Φ )Φ, Φ }
h
h
0 1 2 1 2 2
/2
/2
(19) Finally, the total energy of the beam is obtained as:
∫
∫
+
+
Π
H θ
I u I u w I w J θ u J θ w K θ
I w dx
̇ ]
L
x x xx xx s x x s xx x
s
x
s
L x L L
1
2 0 0,
2
0, 0, 0,2 0, 0, 0, 0,
0,2
0 0,2 0 0, 12 0 0 0,2 0 0
1
2 0 0 0 1 0 0, 2 0,
2
1 0 0 2 0 0, 2 0
2
0 0
(20) Based on the Ritz method[51], the displacement variables in Eq
(20)can be approximated as:
∑
⎛
⎝
⎜
⎞
⎠
⎟=
=
u x t, ψ x u e( )
j
m
x j iωt
0
∑
⎛
⎝
⎜
⎞
⎠
⎟=
=
w x t, φ x w e( )
j
m
j j iωt
0
∑
⎛
⎝
⎜
⎞
⎠
⎟=
=
θ x t, ψ x θ e( )
j
m
x j iωt
0
where ω is the frequency, i2= −1; ( ,u w θ j j, j) are unknown variables;
ψ x j( )andφ x j( )are the shape functions which should be chosen to sa-tisfy the boundary conditions[51] If they do not meet these conditions,
a penalty function method can be applied to recover the boundary conditions[52] Practically however, this approach leads to an increase
in the size of the stiffness and mass matrices and thus causing compu-tational costs Moreover, it is known that the equations of motion of the beams are expressed under the fourth-order differential ones, hence the primary solutions are given in terms of exponential functions It is numerically observed that the approximation offield variables based on the elastic homogeneous solution will lead to an accuracy and fast convergence of the solution Therefore, the present research proposes novel hybrid shape functions given in Table 1 These functions are composed of the exponential functions and admissible trigonometric ones that satisfy automatically various boundary conditions of the beam given inTable 2in which S-S, C-F and C-C are represented for simply supported, clamped-free and clamped-clamped, respectively
The governing equations of motion can be obtained in the form of Lagrange’s equations by substituting Eq (21) into Eq.(20)
∂
∂
d
d
dt d
̇ 0
whered j represents for u w θ( ,j j, j)
Eq.(22)can be written in the matrix forms as:
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎞
⎠
⎟
⎟
⎧
⎨
⎫
⎬=
⎧
⎨
⎫
⎬
θ
ω
u
0
T
T T
T
T T
11 12 13
12 22 23
13 23 33
2
11 12 13
12 22 23
13 23 33
(23)
or under the compact form:
where the components of the stiffness matrixK, mass matrixMand force vectorPare given by:
Table 1 Hybrid shape functions of the Ritz method
S-S
sinπx
L
−
e jx
L
−
e jx L
C-F
( − 1 cosπx L)e−
jx
L sinπx L e−
jx L
C-C
sin πx
L
−
e jx
L
−
e jx L
Table 2 Boundary conditions of the proposed beam model
C-C u0 = 0,w0 = 0,θ0= 0, w0,x= 0 u0 = 0,w0 = 0,θ0= 0, w0,x= 0 C-F u0 = 0,w0 = 0,θ0= 0, w0,x= 0
Trang 4∫ ∫ ∫
∫
∫
∫
=
=
K A ψ ψ dx K B ψ φ dx K B ψ ψ dx
K D φ ψ A φ ψ dx K H ψ ψ A ψ ψ dx
M I ψ ψ dx M I ψ φ dx M J ψ ψ dx
M I φ φ I φ φ dx M J φ ψ dx M
K ψ ψ dx
F qφ dx
ij
L
i x j x ij
L
i x j xx ij s
L
i x j x ij
L
i xx j xx s i x j x i x j x
ij
L s
i xx j x s i x j ij
L s
i x j x s i j ij
L
i j ij
L
i j x ij
L
i j ij
L
i j i x j x ij
L
i x j ij L
i j
i
L
i
11
0 , , 12 0 , , 13 0 , ,
22
0 , , , , 0 , ,
23
11
13
1 0 22
2 0
0
(25)
It is noted from Eq.(24)that the bending responses can be derived
by solving the equationKd=P, the buckling responses by solving the
equation K =0, and vibration responses based on the equation
− ω
K 2M = 0
3 Numerical examples
To demonstrate the accuracy of the proposed beam model, two
shear functions f z( ) involved in Eq (10) are considered:
f z( ) z
h
z
h
3
2
4
3
3
2 [7] (HSBT1) and f z( )=hsin
π πz
h [13] (HSBT2)
Several cases in static, vibration and buckling of laminated composite
beams are carried out to validate the proposed theory The following
materials are used:
•Material I [32]: E2=E G3, 12=G13=0.6E2,G23=
0.5 2, 12 13 23 0.25
•Material II [32]: E2=E G3, 12=G13=0.5E2,G23=
0.2 2, 12 13 23 0.25
For convenience, the following normalized terms are used:
⎝
⎞
⎛
⎝
⎞
⎠
w w E bh
bh
qL σ
L h
σ bh
qL σ
0 2 3
4
2 2
(26a)
=
ω ωL
h
ρ
E
¯
2
=
E bh
cr cr
2
The Ritz method has been widely used for bending, vibration and
buckling analysis of the structures The static responses are related to
the equilibrium problem while the buckling and vibration behaviours
correspond to the eigenvalue ones It is from many previous works that
this method yields upper bound to the exact values for the natural
frequencies and buckling loads, and lower bound for the displacements
in average sense (see Refs.[53–55]for more details) In order to verify
the convergence of the present solutions, a (0°/90°/0°) composite beam
for different boundary conditions with Material I, E E1/ 2 = 40 and
L h/ = 5 has been considered and its results are shown inTable 3 It can
be seen that the present solutions converge when the number of series
m = 12 Therefore, this value will be used herein for the following
numerical examples It is also observed fromTable 3that the solutions
of C-F and C-C beams converge slower than that of S-S beam due to
essential boundary conditions
3.1 Bending analysis
Table 4shows nondimensional mid-span displacements of (0°/90°/
0°) and (0°/90°) composite beams with Material II and E E1/ 2= 25 The
results are reported for S-S, C-C and C-F beams with three different
span-to-thickness ratiosL h/ = 5, 10 and 50 The present solutions are
also compared with those obtained from the HSBTs of Nguyen et al.[4],
Murthy et al.[28], Khdeir and Reddy[56], and the quasi-3D theories of
Nguyen et al.[5], Zenkour [57] It can be observed that all present solutions agree well with existing solutions In addition, the transverse displacements of HSBT1 are closer to those of quasi-3D theory[5,57] than those of the HSBT2, however the HSBTs converge each other when the effect of shear deformation is small (L h/ =50) The comparison of normal stress and shear stress are also introduced inTable 5for S-S beams Again, excellent agreements between these HSBTs are found
To investigate the influence of fibre angle change on the static re-sponses, Table 6 presents the nondimensional transverse mid-span displacements of (0°/θ°/0°) and (0°/θ°) laminated composite beams withL h/ =10 and Material II (E E1/ 2= 25) The results are displayed with variousfibre angles and for different boundary conditions, and then compared with those from the quasi-3D theory of Nguyen et al [5] As expected, the present results comply with those from the pre-vious work The variation of the transverse displacements withfibre angles for S-S beams is illustrated in Fig 2 It can be seen that the deflections of (0°/θ°) beams increase by the increase of the fibre
or-ientation and change rapidly from θ = 5° to θ = 70° Whereas, the
displacements obtained from symmetric beams (0°/θ°/0°) are quite small in comparison with those of asymmetric ones (0°/θ°), which can
be explained by the differences between flexural stiffness of two
lay-Table 3 Convergence studies for (0°/90°/0°) composite beams (L h/ =5, HSBT1, Material I,E E1/ 2=40)
Transverse displacement
Fundamental frequency
Critical buckling load
Table 4 Normalized deflections of (0°/90°/0°) and (0°/90°) beams under uniform loads
(Material II, E E1/ 2= 25)
Trang 53.2 Buckling and free vibration analyses
The comparisons of the normalized fundamental frequencies are
presented inTables 7 and 8, whilst the comparisons of the normalized
critical buckling loads are presented inTables 9 and 10 The
compar-ison is made for different beam configurations, i.e., (0°/90°/0°), (0°/
90°), (0°/θ°/0°) and (0°/θ°) under various span-to-thickness ratios,fibre
orientations and boundary conditions It is noted that although the
bifurcation buckling phenomena does not occur for laminated
compo-site beams with unsymmetric layers, in order to verify the accuracy of
the present beam theory in comparison with the earlier researches, the
critical buckling loads of laminated composite beams with layers (0°/
90°) and (0°/θ°) are still considered The validation of these results are
compared with those derived from the HSBTs ([24,28,4]) and quasi-3D theories ([42,40,5]) It can be seen that the consistencies of the solu-tions are again found
The influence of fibre orientation on the normalized fundamental natural frequencies and critical buckling loads of (0°/θ°/0°) and (0°/θ°) laminated composite beams are displayed inFigs 3 and 4, respectively, for S-S beams It can be seen that the increase offibre angle leads an decrease in the buckling loads and fundamental frequencies The sig-nificant deviations on the results of asymmetric beams are observed
from θ = 5° to θ = 70°, which is the same response with the transverse
displacement
4 Conclusions
A novel unified beam theory has been proposed for the bending, buckling and free vibration analysis of composite beams Based on the fundamental equations of elasticity theory, the displacementfield of the proposed model was derived in a unified form which can be recovered
Table 5
Normalized stresses of (0°/90°/0°) and (0°/90°) S-S beams under uniform loads
(Material II, E E1/ 2= 25)
Table 6
Normalized deflections of [0°/ °θ/0°] and [0°/ °θ ] beams under uniform loads (L h/ = 10, Materials II, E E1/ 2= 25)
Fig 2 Influences of fibre orientation on normalized deflections of ° ° °[0 /θ/0 ] and[0 /°θ°] composite beams under uniform loads (L h/ = 10, Material II,
E E1/ 2= 25)
Trang 6to that of existing HSBTs available in the literature The Ritz method
with a new shape function under hybrid form has been employed to
solve the governing equations of the proposed beam model for stresses,
deflections, buckling loads and natural frequencies A comprehensive
verification study has been conducted, and the numerical results
de-monstrated that the proposed beam model can well predict the bending,
free vibration and buckling responses of composite beams under
dif-ferent boundary conditions and arbitrary lay-ups In addition, the
comprehensive result presented in this paper can also be served as a
reference solution for the development of composite beam models in
the future
CRediT authorship contribution statement TrungKien Nguyen: Conceptualization, Methodology, Writing -original draft Ba-Duy Nguyen: Software, Validation Thuc P Vo: Writing - review & editing.Huu-Tai Thai: Writing - review & editing
Declaration of Competing Interest The authors declare that they have no known competingfinancial interests or personal relationships that could have appeared to influ-ence the work reported in this paper
Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.02-2018.312
Table 7
Normalized fundamental frequencies of (0°/90°/0°) and (0°/90°) composite
beams (Material I, E E1/ 2= 40)
Table 8
Normalized fundamental frequencies of [0°/ °θ/0°] and [0°/ °θ ] composite beams (L h/ = 10, Materials I, E E1/ 2= 40)
Table 9 Normalized buckling loads of (0°/90°/0°) and (0°/90°) composite beams
(Material I, E E1/ 2= 40)
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Table 10
Normalized buckling loads of [0°/ °θ/0°] and [0°/ °θ ] composite beams (L h/ = 10, Materials I, E E1/ 2= 40)
Fig 3 Influences of fibre orientation on normalized fundamental frequencies
of[0 /° θ° °/0 ] and[0 /° θ°] beams (L h/ = 10, Material I, E E1/ 2= 40)
Fig 4 Influences of fibre orientation on normalized critical buckling loads of
° θ° °
[0 / /0 ] and[0 /° θ°] beams (L h/ = 10, Material I, E E1/ 2= 40)
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