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Post-buckling behavior of imperfect laminated composite plates with rotationally-restrained edges
Composite Structures 125 (2015) 117–126
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Post-buckling behavior of imperfect laminated composite plates with rotationally-restrained edges Qingyuan Chen a, Pizhong Qiao a,b,⇑ a b
State Key Laboratory of Ocean Engineering and School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Department of Civil and Environmental Engineering, Washington State University, Sloan Hall 117, Pullman, WA 99164-2910, USA
a r t i c l e
i n f o
Article history: Available online 7 February 2015 Keywords: Nonlinear static analysis Post-buckling Rotationally-restrained laminates Galerkin method In-plane shear loading Combined in-plane loading
a b s t r a c t The nonlinear governing equations of rotationally-restrained laminated composite plates with imperfection are presented by the Galerkin method, and they are solved by employing the Newton–Raphson method for the post-buckling analysis. The considered laminates are symmetric, and they are loaded in pure in-plane shear or combined in-plane shear and compression. The deformation shape function of the restrained plates is obtained through a linear combination of vibration eigenfunctions of simply supported and clamped beams along either the longitudinal or transverse direction of plates. The validity study shows that the presented method is effective for performing the nonlinear analysis of laminates with all four edges elastically-restrained against rotation. A parametric study is conducted to evaluate the effect of rotational spring stiffness, material properties, and fiber orientation under pure in-plane shear as well as the loading ratio under combined shear and compression on the nonlinear static and post-buckling behavior of rotationally-restrained laminates. The proposed solution for nonlinear static analysis of rotationally-restrained composite plates with imperfection is accurate and effective, as demonstrated by the comparisons with the predictions by the finite element analysis, and combined with the discrete plate analysis technique, it can be potentially applied to post-buckling analysis of FRP structural shapes. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The simply supported and clamped boundary conditions are two extreme and ideal cases, and the boundary edges are usually elastically restrained by the adjacent structures in reality. Some researchers have studied the elastic large deflection or post-buckling behavior of isotropic and laminated plates with the edges elastically restrained and under compression by the analytical method [1,2], semi-analytical method [3–5], and numerical method [6,7]. Shear post-buckling of composite laminates has attracted less attention than that of laminates under compression, and most of the existing studies were about the laminates with the simply supported or clamped boundary edges, such as the studies by using the analytical method [8], semi-analytical method [9–14], and numerical method [15,16]. Concerning the laminated plates with elastically-restrained edges subjected to shear loading, Quatmann and Reimerdes [17] presented an analytical method for post-buckling behavior of composite fuselage structures under combined ⇑ Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Mulan Building A903, Shanghai 200240, PR China. E-mail addresses: [email protected], [email protected] (P. Qiao). http://dx.doi.org/10.1016/j.compstruct.2015.01.043 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
compression and shear loading, in which the torsional restraints along the long edges of the plates were considered to simulate the influence of different types of stringers on the post-buckling behavior. Beerhorst et al. [18] also investigated the post-buckling behavior of an infinitely long symmetric and balanced laminate with the longitudinal edges elastically restrained by the torsional springs and under in-plane compression and shear using the analytical method. The above two studies researched the long laminates under shear loading, but the behavior of relatively short laminates under shear loading was not considered. Chia [19] developed a semi-analytical solution for post-buckling analysis of an unsymmetrically-laminated angle-ply rectangular plate under inplane compression and edge shear. In Chia [19]’s study, the opposite edges of the laminates were assumed to be elastically restrained against rotation to the same degree, and the study only presented the numerical results for post-buckling of the square plates under uniaxial and biaxial compression. In this paper, a nonlinear static solution for the relative short imperfect symmetric laminates with four edges rotationally-restrained and subjected to the combined shear and compression (as shown in Fig. 1) is presented. The nonlinear governing equations are derived using the Galerkin method, and the
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Newton–Raphson method is then used to solve nonlinear problem. This method is based on the semi-analytical method proposed by Zhang and Matthews [9,10], Kosteletos [11] and Romeo and Frulla [12] for the non-linear analysis of symmetrically laminated plates with simply-supported or clamped boundary conditions. In the present analysis, the eigenfunctions of simply-supported and clamped beams are linearly combined (or uniquely weighed) to satisfy the rotationally-restrained boundary conditions. The numerical results for relatively short laminates with various rotational spring stiffness under in-plane shear loading are presented and compared with those from the numerical finite element analysis. Then, a parametric study is conducted to examine the effect of a wide range of parameters on the nonlinear static and post-buckling behavior of rotationally-restrained plates under shear or combined shear and compressive loading.
2. Theoretical formulations
The laminated composite plate and coordinate system are shown in Fig. 1, and the length and width of the plate are a and b, respectively. The laminated plate is subjected to the in-plane shear N xy and bi-axial compression N xx and N yy ; in addition, the plate is elastically restrained along all four edges with the rotational spring stiffness k1 at x ¼ 0 and a, and k2 at y ¼ 0 and b, respectively. The laminate considered is thin (the plate thickness h is much smaller than the in-plane dimensions of the plate), so the classical laminated plate theory is used. The constitutive relations for the laminated plate are expressed as:
N M
¼
A
B
B
D
e0 j
T
M xy
2
@ w @x@y 3 D16 7 D26 5 D66
iT
) ð1Þ
where B is called the bending-extension coupling matrix, and it is a zero matrix for the symmetric laminates as considered in this study, and
ð2Þ
in which, N xx ; N yy and N xy are the in-plane normal and shear forces per unit length; M xx ; Myy and Mxy are the bending and twisting moments per unit length; e0xx ; e0yy and e0xy are the normal and shear strains at the middle surface; w is the transverse deflection of every point ðx; yÞ of the middle surface of the plate; Aij ði; j ¼ 1; 2; 6Þ are the in-plane extension stiffness, and Dij ði; j ¼ 1; 2; 6Þ are the bending stiffness (see [20,21]). Partially inverting Eq. (1) and considering only the symmetric laminates lead to
e0 ¼ A1 N M¼Dj
2.1. Governing equation
(
T
N ¼ ½ Nxx Nyy Nxy ; M ¼ ½ M xx M yy h 2 T 2 e0 ¼ e0xx e0yy e0xy ; j ¼ @@xw2 @@yw2 2 3 2 A11 A12 A16 D11 D12 6 7 6 A ¼ 4 A12 A22 A26 5; D ¼ 4 D12 D22 A16 A26 A66 D16 D26
ð3Þ
The equilibrium equations of a generally layered laminate with imperfection under in-plane loading is given as [12,22]:
@Nxx @Nxy þ ¼0 @x @y @Nxy @Nyy þ ¼0 @x @y @ 2 Mxx @ 2 M xy @ 2 Myy @2w @2w @2w þ2 þ þ Nxx 2 þ 2Nxy þ N yy 2 2 2 @x @xy @y @x @xy @y þ N xx
@ 2 w0 @ 2 w0 @ 2 w0 þ 2Nxy þ Nyy ¼0 2 @x @xy @y2
ð4Þ
and the compatibility equation of the laminate with imperfections are reported as [12,22]: 2 2 @ 2 e0xx @ e0yy @ e0xy @ 2 w @ 2 w @ 2 w @ 2 w @ 2 w @ 2 w0 þ þ þ 2 2 @y @x @x@y @x2 @y2 @x@y @x@y @x2 @y2
þ
@ 2 w0 @ 2 w @ 2 w @ 2 w0 2 ¼0 @x@y @x@y @x2 @y2
ð5Þ
Introducing the Airy function /ðx; yÞ:
Nxx ¼
@2/ ; @y2
Nyy ¼
@2/ ; @x2
Nxy ¼
@2/ @x@y
ð6Þ
By substituting Eqs. (6) and (3) into Eqs. (4) and (5), respectively, the first two equilibrium equations in Eq. (4) are satisfied spontaneously, and the following equilibrium and compatibility equations in the dimensionless form are obtained [9–12]:
@4W @n4
þ a1
@4W @n3 @ g 2
þ a2
@4W @n2 @ g2
þ a3
@4W @4W þ a 4 @n@ g3 @ g4
2
@ F @ W @2F @2W @2F @2W @2F @2W 0 2 þ þ 2 2 @ g @n @n@ g @n@ g @n2 @ g2 @ g2 @n2 ! @2F @2W 0 @2F @2W 0 þ 2 ¼0 @n@ g @n@ g @n2 @ g2 a5
@4F @n
4
þ b1
@4F 3
@n @ g 2
þ b2
@4F 2
@n @ g2
þ b3
@4F @4F þ b4 4 3 @n@ g @g
2
@ W @ W @2W @2W @2W @2W 0 þ @n2 @ g2 @n@ g @n@ g @n2 @ g2 ! @2W 0 @2W @2W @2W 0 ¼0 2 þ @n@ g @n@ g @n2 @ g2
þ b5
Fig. 1. Geometry of the rotationally-restrained laminates under combined in-plane shear and compression.
in which, the non-dimensional parameters are defined as
ð7Þ
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w w0 / x y a ; W0 ¼ ; F¼ ; n¼ ; g¼ ; k¼ ; 2 h a b b h A22 h D D ¼ ; A ¼ A22 A1 2 A22 h D ð2D12 þ 4D66 Þ D a1 ¼ 4k 16 a2 ¼ k2 ; a3 ¼ 4k3 26 ; ; D11 D11 D11 D 1 ; a5 ¼ k2 a4 ¼ k4 22 D11 D11 A ð2A12 þ A66 Þ A b1 ¼ 2k 26 b2 ¼ k2 ; b3 ¼ 2k3 16 ; ; A22 A22 A22 A 1 b4 ¼ k4 11 ; b5 ¼ k2 : A22 A22
pressive load, transverse compressive load and in-plane shear load,
W¼
2
respectively. They are defined as ½N 1 N 2 N 12 ¼ ½N xx N yy N xy b 2
=ðA22 h Þ. Considering the rotationally-restrained edges of laminates, the displacement boundary condition and the boundary condition of Airy function are given as follows:
W ¼ 0; W ¼ 0; ð8Þ
In order to satisfy the rotationally-restrained boundary conditions, the approximate deflection function is uniquely constructed by linearly adding the associated weight coefficients xm (or en ) and 1 xm (or 1 en ) to the vibration characteristic functions of the simply-supported and clamped beams, respectively, as 1 X 1 X RR C mn X RR m ðnÞY n ðgÞ
@ F 2
@n
ðM nn Þn¼0 ¼ D11
ð10Þ ð11Þ
gÞ ¼ e
CC n ÞY n ð
gÞ þ ð1 e
gÞ
CC in which, X SS m ðnÞ and X m ðnÞ are the vibration eigenfunctions of simply-supported and clamped beams, respectively, along the x direction:
X SS m ðnÞ ¼ sin mpn CC X CC m ðnÞ ¼ sin am n sinh am n cm ðcos am n cosh am nÞ
cCC m ¼ ðsin am sinh am Þ=ðcos am cosh am Þ; am ¼ 4:7300; 7:8532; . . . Y SS n ð
ð12Þ
Y SS n ðyÞ ¼ sin n Y CC n ðyÞ ¼ sin n CC n ¼ ðsin n
ð13Þ
W 0 ðn; gÞ ¼ I0 X 0 ðnÞY 0 ðgÞ ð14Þ
Y 0 ðgÞ ¼ sinðspgÞ in which, I0 is the amplitude of imperfection, r and s are the numbers of half-wavelength in the longitudinal and transverse directions, respectively. The non-dimensional form of Airy function is given by
F ¼ N 1
g2 2
N2 k2
1 X 1 X n2 CC N12 kng þ vpq X CC p ðnÞY q ðgÞ 2 p¼1 q¼1
2 ðM gg Þg¼0 ¼ 4D22 ðM gg Þg¼1 ¼ 4D22
The initial imperfection is expressed as
X 0 ðnÞ ¼ sinðr pnÞ
" ðM nn Þn¼1 ¼ D11
2
pg a g sinh an g cCC n ðcos an g cosh an gÞ a sinh an Þ=ðcos an cosh an Þ;
ð19Þ
ð20Þ
ð21Þ
@2 W @n2 @2 W @n2
! þ kD16 n¼0
!
@2 W @ g2 @2 W @ g2
þ kD16 n¼1
!
g¼0
! g¼1
1 þ D26 k 1 þ D26 k
!
#
@W @n n¼0 n¼0 ! # 2 @ W @W ¼ ak1 @n@ g @n n¼1 n¼1 ! 3 @2W 5 ¼ bk1 @W @n@ g @ g g¼0 g¼0 ! 3 @2 W 5 ¼ bk2 @W @n@ g @ g g¼1
@2W @n@ g
¼ ak1
ð22Þ
ð23Þ
ð24Þ
ð25Þ
g¼1
For the laminates with only the rotational restraint spring at the boundary edges, it can be assumed that there is no variation about the deflection w ðWÞ with respect to y (g) and x (n) at the opposite longitudinal and transverse boundary edges, respectively. The boundary conditions in Eqs. (22)–(25) of the laminates against rotation at the four edges can be approximated by Eqs. (18)–(21). To make the displacement function Eq. (9) satisfying the boundRR ary conditions represented by Eqs. (18)–(21), W ¼ X RR m ðnÞY n ðgÞ is assumed and substituted into Eqs. (18)–(21). The associated weight coefficients xm can be determined from Eq. (18) or (19), and the associated weight coefficients en can be obtained according to Eq. (20) or (21). The weight coefficients are computed and given as follows:
ð15Þ
CC In the above equation, X CC p ðnÞ and Y q ðgÞ are the vibration eigenfunctions of clamped beam along the x and y directions, respectively. The dimensionless N 1 ; N 2 and N 12 are the applied longitudinal com-
ð18Þ
where, Mnn and Mgg are the non-dimensional form of the moment of Mxx and Myy , respectively. Since D16 – 0 and D26 – 0 are for the symmetrically-laminated anisotropic plates, the boundary conditions of moments at the rotationally-restrained boundary edges can be given by the following equations: "
and gÞ and gÞ are the vibration characteristic functions of simply-supported and clamped beams, respectively, along the y direction:
¼ ak1
g¼1
ðM nn Þn¼0 ¼ D11
Y CC n ð
c an ¼ 4:7300; 7:8532; . . .
ðM gg Þg¼1 ¼ D22
!
@W @n n¼0 @n2 n¼0 ! @2W @W ¼ ak 1 @n n¼1 @n2 n¼1 ! 2 @ W @W ¼ bk 2 @ g2 @ g g¼0 g¼0 ! @2W @W ¼ bk 2 @ g2 @ g g¼1
@2W
ðM gg Þg¼0 ¼ D22
Y RR n ð
ð17Þ
It can be seen that the assumed displacement function and Airy function satisfy the boundary conditions of Eqs. (16) and (17). In addition, the boundary conditions of moment at the longitudinal and transverse opposite edges of the symmetrically-laminated or orthotropic plates with odd half-wavelength in both the directions are expressed:
ðM nn Þn¼1 ¼ D11
SS CC X RR m ðnÞ ¼ xm X m ðnÞ þ ð1 xm ÞX m ðnÞ
@ F ¼ kN12 at g ¼ 0; 1 @n@ g
¼ k2 N2 ;
ð9Þ
where,
ð16Þ
2
m¼1 n¼1
SS nYn ð
@2F ¼ kN12 at n ¼ 0; 1 @n@ g
2
2.2. Deflection and Airy functions
Wðn; gÞ ¼
@2F ¼ N1 ; @ g2
xm ¼
en ¼
2D11 a2m ðsin am sinh am Þ ak1 mpðcos am cosh am Þ þ 2D11 a2m ðsin am sinh am Þ
2D22 a2n ðsin an sinh an Þ bk2 npðcos an cosh an Þ þ 2D22 a2n ðsin an sinh an Þ
ð26Þ ð27Þ
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Finally, the weight coefficients xm and en expressed in Eqs. (26) and (27) are substituted into the displacement function of Eq. (9). It is noted that it will result in two opposite edges simply supported (xm ¼ 1 or en ¼ 1) and two opposite edges clamped (xm ¼ 0 or en ¼ 0) when the rotational spring stiffness k1 or k2 equals to zero and infinite, respectively. 2.3. Solution By employing the Galerkin’s method in Eq. (7), the governing equation can be obtained:
Z
Z
1 0
1 0
" @4W @n4
4
þ a1
@ W @n3 @ g
4
þ a2
@ W @n2 @ g2
4
þ a3
4
@ W @ W þ a4 @n@ g3 @ g4
ð28Þ
By substituting the deflection function Eq. (9) and the Airy function Eq. (15) into the above equations and integrating from 0 to 1 with respect to n and g, the resulting governing equations in the matrix form can be obtained:
Vpq CL U
U
T
Kpq CNL U
Vpq CNL0 U
ð29Þ
¼0
in the above equation, m = 1, 2, . . ., M; n = 1, 2, . . ., N; p = 1, 2, . . ., P; q = 1, 2, . . ., Q; and
U ¼ C 11 C 12 . . . C ðm1ÞNþn . . . C MN1 C MN
h iT J ¼ w111 w112 w1mn w1MN1 w1MN w211 w212 w2pq w2PQ 1 w2PQ
mn T mn T Vmn w1mn ¼ Vmn Kmn EL U ENL þ KENL ENL0 V ELp
pq T pq T w2pq ¼ Vpq Kpq ð34Þ CL U CNL þ KCNL V CNL0 :
The convergence criterion of the nonlinear system is that the ratio of the magnitude of the vector DUn to the magnitude of the vector Unþ1 is smaller than the tolerance error.
ð35Þ
3. Numerical examples
þ b5
mn mn T mn mn Vmn ¼0 EL U U KENL U VENL0 U VELp U con
ð33Þ
the Jacobian matrix of the nonlinear system J can be expressed as
jDUn j 6 error: jUnþ1 j
@2F @2W @2F @2W @2F @2W @2F @2W 0 2 þ þ a5 2 2 @ g @n @n@ g @n@ g @n2 @ g2 @ g2 @n2 !# @2F @2W 0 @2F @2W 0 dWdndg ¼ 0 þ 2 @n@ g @n@ g @n2 @ g2 " Z 1Z 1 4 @ F @4F @4F @4F @4F þ b þ b þ b þ b 1 2 3 4 @n@ g3 @ g4 @n4 @n3 @ g @n2 @ g2 0 0 @2W @2W @2W @2W @2W @2W 0 @2W 0 @2W þ þ @n2 @ g2 @n@ g @n@ g @n2 @ g2 @n2 @ g2 !# @2W @2W 0 dFdndg ¼ 0 2 @n@ g @n@ g
d g1mn T T mn mn T mn T ¼ Vmn ðKmn EL ENL þ KENL ÞU VENL0 V ELp dU d g2pq T pq T pq pq T ¼ Vpq CL ðKCNL þ KCNL ÞU V CNL0 ; dU
v11 v12 . . . vðm1ÞNþn . . . vMN1 vMN
T
ð30Þ
The aforementioned nonlinear static and post-buckling simulation method is programmed in MATLAB. If no otherwise specified, the considered laminates in this study are constructed of the carbon/epoxy T800-3900-2 and the material properties of the lamina are given in [21] as: the longitudinal modulus EL ¼ 155:8 GPa, transverse modulus ET ¼ 8:89 GPa, in-plane shear modulus GLT ¼ 5:14 GPa, and Poisson’s ratio m ¼ 0:3. The width of the laminates is 0.5 m, and the ply thickness of the lamina is 0.125 mm. xx ; N yy and The nondimensional normal force coefficients N xy in the following are given by shear force coefficient N
2 xx ; N yy ; N xy ¼ Nxx ; Nyy ; Nxy b N 3 E2 h
ð36Þ
The non-dimensional longitudinal and transverse rotational spring stiffness K 1 and K 2 are used in the present study for the nonlinear static and post-buckling behavior of composite plates, and they are presented as:
K1 ¼
k1 a ; D11
K2 ¼
k2 b D22
ð37Þ
Kpq CNL Þ are shown in Appendix with all the terms being calculated numerically. The set of nonlinear equations can be written in a single expression as:
For the sake of convenience, the dimensionless parameters K ¼ K 1 ¼ K 2 are introduced if the non-dimensional longitudinal rotational spring stiffness K 1 equals the non-dimensional transverse rotational spring stiffness K 2 . The relatively short laminates are considered, i.e., it is assumed that there is only one half-wavelength in both the directions in the post-buckling mode of the laminate. The initial imperfection function with r ¼ s = 1 is introduced as:
h iT f ¼ g111 g112 g1mn ... g1MN1 g1MN g211 g212 ... g2mn g2MN1 g2MN
W 0 ðn; gÞ ¼ I0 sinðpnÞ sinðpgÞ
mn
and the detail of the constant ðcon Þ, vectors mn mn mn pq pq pq mn (Vmn EL ; VENL ; VENL0 ; V ELp ; VCL ; VCNL and VCNL0 ) and matrices ðKENL and
¼ ½0ðMNþPQÞ1 mn
mn mn T mn mn g1 ¼ Vmn EL U U KENL U V ENL0 U V ELp U con pq pq pq T pq g2 ¼ VCL U U KCNL U VCNL0 U
ð31Þ
and it can be solved by the Newton–Raphson method. It is assumed that the laminate is gradually (step-by-step) loaded with the applied load from zero to the final one. First, set the initial guess solution U1 ¼ ½0ðMNþPQ Þ1 , and then obtain the successive approximations to the solution from
Unþ1 ¼ Un J1 f ¼ Un þ DUn
ð38Þ
ð32Þ
with DUn ¼ Unþ1 Un ¼ J1 f, where J is the Jacobian matrix of the nonlinear system. Because the derivative of g1mn and g2pq in Eq. (30) with respect to the vector U are given as follows
The dimensionless central deflection W of the laminates is the maximum deflection, and it should contain the central imperfection, i.e.,
W ¼ Wð0:5; 0:5Þ þ W 0 ð0:5; 0:5Þ:
ð39Þ
3.1. Convergence and validity studies The results of the present method are compared with those obtained by employing the finite element method (FEM) in this section. The finite element analysis is performed by applying the commercial software ABAQUS; the element size of 0:0125 0:0125 m is used in FEM, and the shell element S4R is considered. First, the restrained square symmetrically-laminated orthotropic plates (RSSLOP) [0°/90°/0°/90°/0°/0°/0°/90°]S with all the edges
Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Fig. 2. Convergence study for the load–deflection curves of RSSLOP with K ¼ 10 under pure in-plane shear.
rotationally restrained under pure in-plane shear loading are studied. The load–deflection curves with various approximation terms (i.e., M ¼ N ¼ P ¼ Q ¼ 3, 4, 5 and 6) for the shear post-buckling behavior of the RSSLOP with the dimensionless rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 10 are shown in Fig. 2, and the results are compared with those from FEM. As shown in Fig. 2, the results based on the approximation terms of 4–6 are almost identical, and the number of terms M ¼ N ¼ P ¼ Q ¼ 5 is thus used in the following nonlinear static and post-buckling analysis for RSSLOP. It is noted that the amplitude of the imperfection I0 ¼ 0:01 is selected in the analysis. In order to determine the effect of the amplitude of the imperfection I0 on the in-plane shear nonlinear static and post-buckling behavior of the RSSLOP with the restrained rotational stiffness K ¼ 10, different amplitudes of the imperfection are investigated in Fig. 3. It can be seen that the amplitude of imperfection I0 only has a relatively large influence on the early stage of the load–deflection curves. It can be expected that there is only a slight difference of the load–deflection patterns of laminates with very small amplitude of imperfection I0 and no imperfection. Thus, the amplitude of the imperfection I0 ¼ 0:01 is then selected in the following study in view of the stability in the convergence and accuracy of the results.
Fig. 3. Effect of the amplitude of imperfection I0 on the post-buckling behavior of RSSLOP with K ¼ 10 under pure in-plane shear.
121
Fig. 4. Post-buckling behavior of RSSLOP with various K values.
Fig. 5. Post-buckling behavior of RSSLOP with various K 1 values and constant K 2 ¼ 10.
The nonlinear static and post-buckling behavior of RSSLOP with various rotationally-restrained coefficient K values is examined in Fig. 4; while the nonlinear static and post-buckling behavior of RSSLOP with various dimensionless longitudinal rotational spring stiffness K 1 values and constant dimensionless transverse rotational spring stiffness K 2 ¼ 10 is evaluated in Fig. 5. It can be seen that the results agree quite well with those obtained from FEM, and the laminate will become stiffer gradually with the increase in the dimensionless rotational spring stiffness K (Fig. 4) or dimensionless longitudinal rotational spring stiffness K 1 (Fig. 5). The restrained square symmetrically-laminated anisotropic plates (RSSLAP) [45°/45°/45°/45°/45°/45°/0°/0°]S with all the boundary edges restrained against rotation and with imperfection amplitude of I0 ¼ 0:01 are considered. The convergence study of the nonlinear static and post-buckling response of RSSLAP with the dimensionless rotational spring stiffness of K ¼ K 1 ¼ K 2 ¼ 10 and 60 is given in Figs. 6 and 7, respectively. As shown in Figs. 6 and 7, there is a little discrepancy between the load-defection curves based on the approximation terms of M ¼ N ¼ P ¼ Q ¼ 4, 5 and 6 from Figs. 6 and 7, which is consistent with the results given in Fig. 2. Thus, the number of terms M ¼ N ¼ P ¼ Q ¼ 5 is chosen with confidence for the nonlinear static and post-buckling analysis in the following study. The load–deflection curves of RSSLAP with
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
(a) Negative shear
(a) Negative shear
(b) Positive shear
(b) Positive shear
Fig. 6. Convergence study for the load deflection curves of RSSLAP with K ¼ 10 and I0 ¼ 0:01 under pure in-plane shear.
Fig. 7. Convergence study for the load deflection curves of RSSLAP with K ¼ 60 and I0 ¼ 0:01 under pure in-plane shear.
the rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 0; 10; 20; 40; 60; 1 under the positive and negative in-plane shear loading are shown in Fig. 8, and it can be seen that the present method can be used to investigate the nonlinear static and post-buckling behavior of the symmetric laminates, although the large rotational spring stiffness K (i.e., K ¼ 40; 60; 1) results in some relative discrepancies in the large deformation regime between the present analytical solution and FEM. In order to illustrate the computational efficiency and cost of applying the present semi-analytical method, the required computing time of the nonlinear static and post-buckling problem of symmetric laminates is shown in Table 1. The results are obtained using the workstation (Intel Xeon X5675 3.07 GHz processor, 64 GB). Because the computing time mainly relates to the approximation terms of displacement function and Airy function and the chosen number of the approximation terms M ¼ N ¼ P ¼ Q ¼ 5 is considered in this study for the nonlinear static and post-buckling analysis of both the symmetricallylaminated orthotropic and anisotropic plates as shown in the convergence study, the computational time (CPU) for the restrained square symmetrically laminated orthotropic plate (RSSLOP) [0°/ 90°/0°/90°/0°/0°/0°/90°]S with dimensionless rotational spring
stiffness K ¼ K 1 ¼ K 2 ¼ 10 are compared in Table 1 between the present semi-analytical method and FEM. As shown in Table 1, the results demonstrate that the computing efficiency of the present semi-analytical approach is dramatically improved when compared to the analysis using FEM. 3.2. Parametric study of the imperfect laminates under pure in-plane shear The effect of rotational restraint stiffness on the nonlinear static and post-buckling response of the orthotropic and symmetric laminates is examined in the above section, and the rotational spring stiffness have a major impact on the post-buckling stiffness of the laminates; i.e., it is easier to produce a large deflection for the laminates with smaller rotational restraint stiffness, and a small increment of shear is required for a given increment of central deflection for the laminates edged with a small rotational restraint stiffness. A laminate is a complex material that is often made of multiple layers of a particular lamina with different fiber orientations. Through investigating the nonlinear static and post-buckling behavior of the single lamina, the full understanding of the nonlinear static
Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
(a) Negative shear
123
Fig. 9. Effect of the orthotropy parameter a ¼ EL =ET on the post-buckling behavior of RSLSOP under pure in-plane shear.
(b) Positive shear Fig. 8. Validity study for post-buckling of RSSLAP with various K values and I0 ¼ 0:01 under pure in-plane shear.
Fig. 10. Effect of in-plane shear modulus GLT on the post-buckling behavior of RSLSOP under pure in-plane shear.
Table 1 CPU time for post-buckling analysis of RSSLOP.
CPU time (s)
Present semi-analytical method
FEM
5.50
80.50
response of the whole laminate can be manifested. First, the effect of material properties of the lamina on the nonlinear static and postbuckling behavior of single layer orthotropic plates is investigated. Then, the effect of fiber orientation on the nonlinear static and post-buckling behavior of single layer anisotropic plates is studied. It is noted that the thickness of the considered single layer square plates with a ¼ b = 0.5 is set as h = 0.002 m. The results presented in Fig. 9 show the effect of the orthotropy parameter a ¼ EL =ET on the nonlinear static and post-buckling behavior of the restrained single layer square orthotropic plates (RSLSOP) with the rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 10 and imperfection of I0 ¼ 0:01 under pure shear. The material properties of RSLSOP are: ET ¼ 8:89 GPa, GLT ¼ 5:14 GPa and mLT ¼ 0:3. As shown in Fig. 9, there is almost no difference on their post-buckling stiffness for RSLSOP with various orthotropy parameter a when under the pure in-plane shear action. The effect of the shear
Fig. 11. Effect of in-plane Poisson’s ratio RSLSOP under pure shear.
mLT on the post-buckling behavior of
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
modulus GLT on the nonlinear static and post-buckling response of the RSLSOP with the parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET and mLT ¼ 0:3 under pure in-plane shear is shown in Fig. 10. As shown in Fig. 10, the post-buckling stiffness of the orthotropic plate increases with the increasing shear modulus. While for the examples shown in Fig. 11, with the fixed parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa; EL ¼ 6ET and GLT ¼ 0:5ET , the effect of the in-plane Poisson’s ratio of RSLSOP is illustrated. As shown in Fig. 11, the variation of mLT does not make much influence on the post-buckling stiffness of RSLSOP. The effect of fiber orientation angle h on the nonlinear static and post-buckling behavior of restrained single layer square anisotropic plates (RSLSAP) [h] with the parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET ; GLT ¼ 0:5ET and mLT ¼ 0:3 under pure in-plane shear is examined (see Fig. 12). As shown in Fig. 12, the smaller the fiber orientation angle h, the larger the differences of the nonlinear static and post-buckling behavior between the applied positive and negative shear. The effect of the fiber orientation angle h on the nonlinear static and post-buckling response of the restrained square symmetrically laminated anisotropic plate (RSSLAP) ½h= h4S with parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET ; GLT ¼ 0:5ET and mLT ¼ 0:3 under pure in-plane shear is shown in Fig. 13. As depicted in Fig. 13, the nonlinear static behavior approaches to the one of orthotropic laminates when the number of layers (4 m) of RSSLAP ½h= hmS increases. As observed in Fig. 12, when the fiber orientation angle h increases (from the minimum h ¼ 0 up to the maximum of h ¼ 45 ), the postbuckling stiffness of RSLSAP under the positive in-plane shear will also increase; whereas the fiber orientation angle h shows less of an effect on the post-buckling stiffness of the RSLSAP under the negative in-plane shear. As expected, the post-buckling stiffness of RSSLAP ½h= h4S is proportional to the fiber orientation angle h (up to h ¼ 45 ) when the laminates are subjected to the positive or negative in-plane shear, which can also be obtained from Fig. 13.
Fig. 13. Effect of fiber orientation h on the post-buckling behavior of RSSLAP ½h= h4S under pure in-plane shear.
3.3. Parametric study for laminates under combined shear and compression The effect of loading ratio on the nonlinear static and postbuckling behavior of restrained square symmetrically laminated anisotropic plate (RSSLAP) ½45= 454S with the rotational restraint stiffness K ¼ 10 under combined shear and uniaxial compression or combined shear and biaxial compression loading is studied in Figs. 14 and 15, respectively. As shown in Figs. 14 and 15, the
Fig. 12. Effect of the fiber orientation angle h on the post-buckling behavior of RSLSAP ½h under pure in-plane shear.
Fig. 14. Effect of loading ratio on the post-buckling behavior of RSSLAP ½45= 454S under combined shear and uniaxial compression.
Fig. 15. Effect of loading ratio on the post-buckling behavior of RSSLAP ½45= 454S under combined shear and biaxial compression.
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post-buckling stiffness of RSSLAP decreases with the increasing of the shear to compression loading ratios. 4. Conclusions In this study, the nonlinear governing equations for the nonlinear static and post-buckling behavior of imperfect symmetric laminates with all the edges elastically restrained against rotation and under pure in-plane shear and combined shear and compression are derived using the Galerkin method, and the Newton–Raphson method is employed to solve the nonlinear problem. The displacement functions are uniquely chosen by linearly combining the vibration eigenfunctions of simply supported and clamped beams to meet the rotationally-restrained boundary conditions in either the x- or y-direction of plate. As shown in validity study, the calculated results compare well with those obtained from FEM for both the orthotropic and symmetric laminates; while there are some small discrepancies between the load–deflection curves of the present method and FEM when the dimensionless rotational restraint stiffness becomes larger. A parametric study on the effect of a wide range of parameters to the nonlinear static and post-buckling behavior of restrained laminated plates is conducted, and the following conclusions are reached from the parametric study: (1) When under in-plane shear loading, the post-buckling stiffness of laminates increases with the shear modulus GLT ; while the orthotropic parameter EL =ET and Poisson’s ratio mLT do not much affect the post-buckling stiffness of laminates. (2) The fiber orientation angle h has a significant influence on the post-buckling stiffness of laminates under positive inplane shear, and the post-buckling stiffness of the symmetric laminates ½h= h4S under positive or negative in-plane shear is proportional to the fiber orientation angle h (up to h ¼ 45 ) when the fiber orientation angle increases from the minimum h ¼ 0 up to the maximum of h ¼ 45 . (3) The post-buckling stiffness of the laminates decreases with the increasing ratio of the shear to compression loading (for both the combined shear-uniaxial compression and combined shear-biaxial compression) when the plate is under the combined loading. In summary, a semi-analytical solution for the nonlinear static and post-buckling behavior of rotationally-restrained laminated plates with imperfection based on the Galerkin’s method is obtained, and it can efficiently and accurately perform the nonlinear static and post-buckling analysis as demonstrated by the comparisons with the results from the finite element analysis. Combined with the discrete plate analysis technique, the present solution can be potentially applied to nonlinear static and postbuckling analysis of FRP structural shapes, as similarly shown in a recent study using the numerical spline finite strip method by the authors [7]. Acknowledgment This research is partially supported by the National Natural Science Foundation of China (No. 51478265), and their financial support is gratefully acknowledged. Appendix A mn mn The detail of the constant ðconmn Þ, vectors (Vmn EL ; VENL ; VENL0 ; pq pq pq mn pq VCL ; VCNL and VCNL0 Þ and matrices (KENL and KCNL ) in governing equations are:
Vmn ELp ;
mn Vmn V2EL EL ¼ V1EL h i mn mn V1EL ¼ dði1ÞNþj ; V2EL ¼ ½01PQ 1MN Z 1 Z 1 " 4 RR @Y RR @ X i ðnÞ RR @ 3 X RR mn j ðgÞ i ðnÞ dði1ÞNþj ¼ Y ð g Þ þ a 1 j 4 3 g @ @n @n 0 0
@ 2 Y RR @ 3 Y RR @ 2 X RR @X RR j ðgÞ j ðgÞ i ðnÞ i ðnÞ þ a 3 2 2 @n g g3 @ @ @n # 4 RR @ Y j ðgÞ RR X m ðnÞY RR þ a4 X RR i ðnÞ n ðgÞdndg @ g4 þ a2
" Kmn ENL
¼
K11ENL
K12ENL
K21mn ENL
K22ENL
ðA:1Þ
# ;
K11ENL ¼ ½0MNMN ; K12ENL ¼ ½0MNPQ ; K22ENL ¼ ½0PQPQ ; h i mn K21mn ENL ¼ g ðp1ÞQ þq;ði1ÞNþj PQ MN Z 1 Z 1" 2 RR @ 2 Y CC q ðgÞ @ X i ðnÞ RR CC g mn ¼ a X ðnÞ Y j ðgÞ 5 ðp1ÞQþq;ði1ÞNþj p 2 @ g @n2 0 0 CC RR RR @X CC p ðnÞ @Y q ðgÞ @X i ðnÞ @Y j ðgÞ @n @n @g @g # 2 CC 2 RR @ X p ðnÞ CC @ Y j ðgÞ RR RR þ Y ð g ÞX ðnÞ X m ðnÞY RR q i n ðgÞdndg ðA:2Þ @ g2 @n2
2
mn Vmn V2ENL0 ENL0 ¼ V1ENL0 h i mn ; V2ENL0 ¼ ½01PQ V1mn ENL0 ¼ lði1ÞNþj 1MN Z 1 Z 1" 2 @ 2 Y CC mn j ðgÞ @ X 0 ðnÞ lði1ÞNþj ¼ a5 I0 X CC Y 0 ðgÞ i ðnÞ 2 @g @n2 0 0 @Y CC @X CC j ðgÞ @X 0 ðnÞ @Y 0 ðgÞ i ðnÞ @n @n @g @g # 2 CC 2 @ X i ðnÞ CC @ Y 0 ðgÞ RR þ Y ð g ÞX ðnÞ X m ðnÞY RR 0 j n ðgÞdndg @ g2 @n2 2
ðA:3Þ
mn V2ELp Vmn ELp ¼ V1ELp h i mn ; V2EL ¼ ½01PQ V1mn ELp ¼ eði1ÞNþj 1MN Z 1 Z 1" @Y RR @ 2 X RR @X RR j ðgÞ i ðnÞ RR i ðnÞ emn N1 Y ð g Þ þ 2kN 12 j ði1ÞNþj ¼ a5 2 @n g @ @n 0 0 # 2 RR @ Y j ðgÞ RR þ k2 N2 X RR ðA:4Þ X m ðnÞY RR i ðnÞ n ðgÞdndg @ g2 Z
1
Z
1
"
@ 2 X 0 ðnÞ
@X 0 ðnÞ @Y 0 ðgÞ Y 0 ðgÞ þ 2kN12 @n @g @n2 # 2 @ Y 0 ðgÞ RR ðA:5Þ X m ðnÞY RR þ k2 N2 X 0 ðnÞ n ðgÞ dndg @ g2
conmn ¼ a5 I0
0
Vpq CL ¼ V1CL
N1
0
V2pq CL
h i pq V2pq CL ¼ qði1ÞNþj 1PQ Z 1 " 4 CC @Y CC @ X i ðnÞ CC @ 3 X CC j ðgÞ i ðnÞ Y ð g Þ þ b 1 j 4 3 g @ @n @n 0
V1CL ¼ ½01MN ; qpq ði1ÞNþj ¼
Z
1
0
@ 2 Y CC @ 3 Y CC @ 2 X CC @X CC j ðgÞ j ðgÞ i ðnÞ i ðnÞ þ b 3 2 2 @n g g3 @ @ @n # 4 CC @ Y j ðgÞ CC X p ðnÞY CC þ b4 X CC i ðnÞ q ðgÞdndg @ g4
þ b2
ðA:6Þ
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
" Kpq CNL ¼
K11pq CNL
K12CNL
K21CNL
K22CNL
# ;
K12ENL ¼ ½0MNPQ ; K21ENL ¼ ½0PQMN ; K22ENL ¼ ½0PQPQ ; h i pq K11pq ENL ¼ hðm1ÞNþn;ði1ÞNþj MNMN Z 1 Z 1 " RR RR @Y RR @X m ðnÞ @Y RR pq j ðgÞ n ðgÞ @X i ðnÞ hðm1ÞNþn;ði1ÞNþj ¼ b5 @n @ @n g g @ 0 0 # 2 RR @ 2 Y RR CC n ðgÞ @ X i ðnÞ RR Y j ðgÞ X CC ðA:7Þ X RR m ðnÞ p ðnÞY q ðgÞdndg 2 @g @n2 Vpq CNL0 ¼ V1CNL0
V2pq CNL0
h i pq V2pq CNL0 ¼ sði1ÞNþj 1PQ Z 1Z 1" 2 RR @ Y ð g Þ @ 2 X 0 ðnÞ j ¼ b5 I 0 X RR Y 0 ðgÞ i ðnÞ @ g2 @n2 0 0
V1CNL0 ¼ ½01MN ; spq ði1ÞNþj
@Y RR @X RR j ðgÞ @X 0 ðnÞ @Y 0 ðgÞ i ðnÞ @n @n @g @g # 2 RR 2 @ X i ðnÞ RR @ Y 0 ðgÞ CC Y ð g ÞX ðnÞ X p ðnÞY CC 0 j q ðgÞdndg: @ g2 @n2
þ2
ðA:8Þ
References [1] Paik JK, Kim DK, Lee H, Shim YL. A method for analyzing elastic large deflection behavior of perfect and imperfect plates with partially rotation-restrained edges. J Offshore Mech Arct Eng 2012:134. [2] Vescovini R, Bisagni C. Single-mode solution for post-buckling analysis of composite panels with elastic restraints loaded in compression. Compos Part B Eng 2012;43:1258–74. [3] Rhodes J, Harvey JM. The post-buckling behaviour of thin flat plates in compression with the unloaded edges elastically restrained against rotation. J Mech Eng Sci 1971;13:82–91. [4] Bisagni C, Vescovini R. Analytical formulation for local buckling and postbuckling analysis of stiffened laminated panels. Thin-Walled Struct 2009;47:318–34.
[5] Stamatelos DG, Labeas GN, Tserpes KI. Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels. ThinWalled Struct 2011;49:422–30. [6] Khong PW, Ong JH. Post buckling of a plate with elastic restraints at two unloaded edges. Comput Struct 1995;56:933–8. [7] Qiao P, Chen Q. Post-local-buckling of fiber-reinforced plastic composite structural shapes using discrete plate analysis. Thin-Walled Struct 2014;84:68–77. [8] Mittelstedt C, Erdmann H, Schroder KU. Postbuckling of imperfect rectangular composite plates under inplane shear closed-form approximate solutions. Arch Appl Mech 2011;81:1409–26. [9] Matthews FL, Zhang Y. Postbuckling behavior of anisotropic laminated plates under pure shear and shear combined with compressive loading. AIAA J 1984;22:281–6. [10] Zhang Y, Matthews FL. Large deflection behavior of simply supported laminated panels under in-plane loading. J Appl Mech 1985;52:553–8. [11] Kosteletos S. Postbuckling response of laminated plates under shear load. Compos Struct 1992;20:137–45. [12] Romeo G, Frulla G. Nonlinear analysis of anisotropic plates with initial imperfections and various boundary conditions subjected to combined biaxial compression and shear loads. Int J Solids Struct 1994;31:763–83. [13] Stein M. Postbuckling of long orthotropic plates in combined shear and compression. AIAA J 1985;23:788–94. [14] Stein M. Postbuckling of long orthotropic plates under combined loading. AIAA J 1985;23:1267–72. [15] Wang S, Dawe DJ. Spline FSM postbuckling analysis of shear-deformable rectangular laminates. Thin-Walled Struct 1999;34:163–78. [16] Chen QY, Qiao PZ. Post-buckling analysis of composite plates under combined compression and shear loading using finite strip method. Finite Elem Anal Des 2014;83:33–42. [17] Quatmann M, Reimerdes H-G. Preliminary design of composite fuselage structures using analytical rapid sizing methods. CEAS Aeronaut J 2011;2:231–41. [18] Beerhorst M, Seibel M, Mittelstedt C. Fast analytical method describing the postbuckling behavior of long, symmetric, balanced laminated composite plates under biaxial compression and shear. Compos Struct 2012;94:2001–9. [19] Chia CY. Postbuckling of composite plates under inplane compressive and shear loading having edges elastically restrained against rotation. Composite structures, vol. 3. Springer; 1985. p. 278–89. [20] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton, Florida: CRC Press; 2003. [21] Barbero EJ. Introduction to composite materials design. Boca Raton, Florida: CRC Press; 2010. [22] Chia CY. Nonlinear analysis of plates. New York, NY: McGraw-Hill; 1980.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Post-buckling behavior of imperfect laminated composite plates with rotationally-restrained edges Qingyuan Chen a, Pizhong Qiao a,b,⇑ a b
State Key Laboratory of Ocean Engineering and School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Department of Civil and Environmental Engineering, Washington State University, Sloan Hall 117, Pullman, WA 99164-2910, USA
a r t i c l e
i n f o
Article history: Available online 7 February 2015 Keywords: Nonlinear static analysis Post-buckling Rotationally-restrained laminates Galerkin method In-plane shear loading Combined in-plane loading
a b s t r a c t The nonlinear governing equations of rotationally-restrained laminated composite plates with imperfection are presented by the Galerkin method, and they are solved by employing the Newton–Raphson method for the post-buckling analysis. The considered laminates are symmetric, and they are loaded in pure in-plane shear or combined in-plane shear and compression. The deformation shape function of the restrained plates is obtained through a linear combination of vibration eigenfunctions of simply supported and clamped beams along either the longitudinal or transverse direction of plates. The validity study shows that the presented method is effective for performing the nonlinear analysis of laminates with all four edges elastically-restrained against rotation. A parametric study is conducted to evaluate the effect of rotational spring stiffness, material properties, and fiber orientation under pure in-plane shear as well as the loading ratio under combined shear and compression on the nonlinear static and post-buckling behavior of rotationally-restrained laminates. The proposed solution for nonlinear static analysis of rotationally-restrained composite plates with imperfection is accurate and effective, as demonstrated by the comparisons with the predictions by the finite element analysis, and combined with the discrete plate analysis technique, it can be potentially applied to post-buckling analysis of FRP structural shapes. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The simply supported and clamped boundary conditions are two extreme and ideal cases, and the boundary edges are usually elastically restrained by the adjacent structures in reality. Some researchers have studied the elastic large deflection or post-buckling behavior of isotropic and laminated plates with the edges elastically restrained and under compression by the analytical method [1,2], semi-analytical method [3–5], and numerical method [6,7]. Shear post-buckling of composite laminates has attracted less attention than that of laminates under compression, and most of the existing studies were about the laminates with the simply supported or clamped boundary edges, such as the studies by using the analytical method [8], semi-analytical method [9–14], and numerical method [15,16]. Concerning the laminated plates with elastically-restrained edges subjected to shear loading, Quatmann and Reimerdes [17] presented an analytical method for post-buckling behavior of composite fuselage structures under combined ⇑ Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Mulan Building A903, Shanghai 200240, PR China. E-mail addresses: [email protected], [email protected] (P. Qiao). http://dx.doi.org/10.1016/j.compstruct.2015.01.043 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
compression and shear loading, in which the torsional restraints along the long edges of the plates were considered to simulate the influence of different types of stringers on the post-buckling behavior. Beerhorst et al. [18] also investigated the post-buckling behavior of an infinitely long symmetric and balanced laminate with the longitudinal edges elastically restrained by the torsional springs and under in-plane compression and shear using the analytical method. The above two studies researched the long laminates under shear loading, but the behavior of relatively short laminates under shear loading was not considered. Chia [19] developed a semi-analytical solution for post-buckling analysis of an unsymmetrically-laminated angle-ply rectangular plate under inplane compression and edge shear. In Chia [19]’s study, the opposite edges of the laminates were assumed to be elastically restrained against rotation to the same degree, and the study only presented the numerical results for post-buckling of the square plates under uniaxial and biaxial compression. In this paper, a nonlinear static solution for the relative short imperfect symmetric laminates with four edges rotationally-restrained and subjected to the combined shear and compression (as shown in Fig. 1) is presented. The nonlinear governing equations are derived using the Galerkin method, and the
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Newton–Raphson method is then used to solve nonlinear problem. This method is based on the semi-analytical method proposed by Zhang and Matthews [9,10], Kosteletos [11] and Romeo and Frulla [12] for the non-linear analysis of symmetrically laminated plates with simply-supported or clamped boundary conditions. In the present analysis, the eigenfunctions of simply-supported and clamped beams are linearly combined (or uniquely weighed) to satisfy the rotationally-restrained boundary conditions. The numerical results for relatively short laminates with various rotational spring stiffness under in-plane shear loading are presented and compared with those from the numerical finite element analysis. Then, a parametric study is conducted to examine the effect of a wide range of parameters on the nonlinear static and post-buckling behavior of rotationally-restrained plates under shear or combined shear and compressive loading.
2. Theoretical formulations
The laminated composite plate and coordinate system are shown in Fig. 1, and the length and width of the plate are a and b, respectively. The laminated plate is subjected to the in-plane shear N xy and bi-axial compression N xx and N yy ; in addition, the plate is elastically restrained along all four edges with the rotational spring stiffness k1 at x ¼ 0 and a, and k2 at y ¼ 0 and b, respectively. The laminate considered is thin (the plate thickness h is much smaller than the in-plane dimensions of the plate), so the classical laminated plate theory is used. The constitutive relations for the laminated plate are expressed as:
N M
¼
A
B
B
D
e0 j
T
M xy
2
@ w @x@y 3 D16 7 D26 5 D66
iT
) ð1Þ
where B is called the bending-extension coupling matrix, and it is a zero matrix for the symmetric laminates as considered in this study, and
ð2Þ
in which, N xx ; N yy and N xy are the in-plane normal and shear forces per unit length; M xx ; Myy and Mxy are the bending and twisting moments per unit length; e0xx ; e0yy and e0xy are the normal and shear strains at the middle surface; w is the transverse deflection of every point ðx; yÞ of the middle surface of the plate; Aij ði; j ¼ 1; 2; 6Þ are the in-plane extension stiffness, and Dij ði; j ¼ 1; 2; 6Þ are the bending stiffness (see [20,21]). Partially inverting Eq. (1) and considering only the symmetric laminates lead to
e0 ¼ A1 N M¼Dj
2.1. Governing equation
(
T
N ¼ ½ Nxx Nyy Nxy ; M ¼ ½ M xx M yy h 2 T 2 e0 ¼ e0xx e0yy e0xy ; j ¼ @@xw2 @@yw2 2 3 2 A11 A12 A16 D11 D12 6 7 6 A ¼ 4 A12 A22 A26 5; D ¼ 4 D12 D22 A16 A26 A66 D16 D26
ð3Þ
The equilibrium equations of a generally layered laminate with imperfection under in-plane loading is given as [12,22]:
@Nxx @Nxy þ ¼0 @x @y @Nxy @Nyy þ ¼0 @x @y @ 2 Mxx @ 2 M xy @ 2 Myy @2w @2w @2w þ2 þ þ Nxx 2 þ 2Nxy þ N yy 2 2 2 @x @xy @y @x @xy @y þ N xx
@ 2 w0 @ 2 w0 @ 2 w0 þ 2Nxy þ Nyy ¼0 2 @x @xy @y2
ð4Þ
and the compatibility equation of the laminate with imperfections are reported as [12,22]: 2 2 @ 2 e0xx @ e0yy @ e0xy @ 2 w @ 2 w @ 2 w @ 2 w @ 2 w @ 2 w0 þ þ þ 2 2 @y @x @x@y @x2 @y2 @x@y @x@y @x2 @y2
þ
@ 2 w0 @ 2 w @ 2 w @ 2 w0 2 ¼0 @x@y @x@y @x2 @y2
ð5Þ
Introducing the Airy function /ðx; yÞ:
Nxx ¼
@2/ ; @y2
Nyy ¼
@2/ ; @x2
Nxy ¼
@2/ @x@y
ð6Þ
By substituting Eqs. (6) and (3) into Eqs. (4) and (5), respectively, the first two equilibrium equations in Eq. (4) are satisfied spontaneously, and the following equilibrium and compatibility equations in the dimensionless form are obtained [9–12]:
@4W @n4
þ a1
@4W @n3 @ g 2
þ a2
@4W @n2 @ g2
þ a3
@4W @4W þ a 4 @n@ g3 @ g4
2
@ F @ W @2F @2W @2F @2W @2F @2W 0 2 þ þ 2 2 @ g @n @n@ g @n@ g @n2 @ g2 @ g2 @n2 ! @2F @2W 0 @2F @2W 0 þ 2 ¼0 @n@ g @n@ g @n2 @ g2 a5
@4F @n
4
þ b1
@4F 3
@n @ g 2
þ b2
@4F 2
@n @ g2
þ b3
@4F @4F þ b4 4 3 @n@ g @g
2
@ W @ W @2W @2W @2W @2W 0 þ @n2 @ g2 @n@ g @n@ g @n2 @ g2 ! @2W 0 @2W @2W @2W 0 ¼0 2 þ @n@ g @n@ g @n2 @ g2
þ b5
Fig. 1. Geometry of the rotationally-restrained laminates under combined in-plane shear and compression.
in which, the non-dimensional parameters are defined as
ð7Þ
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
w w0 / x y a ; W0 ¼ ; F¼ ; n¼ ; g¼ ; k¼ ; 2 h a b b h A22 h D D ¼ ; A ¼ A22 A1 2 A22 h D ð2D12 þ 4D66 Þ D a1 ¼ 4k 16 a2 ¼ k2 ; a3 ¼ 4k3 26 ; ; D11 D11 D11 D 1 ; a5 ¼ k2 a4 ¼ k4 22 D11 D11 A ð2A12 þ A66 Þ A b1 ¼ 2k 26 b2 ¼ k2 ; b3 ¼ 2k3 16 ; ; A22 A22 A22 A 1 b4 ¼ k4 11 ; b5 ¼ k2 : A22 A22
pressive load, transverse compressive load and in-plane shear load,
W¼
2
respectively. They are defined as ½N 1 N 2 N 12 ¼ ½N xx N yy N xy b 2
=ðA22 h Þ. Considering the rotationally-restrained edges of laminates, the displacement boundary condition and the boundary condition of Airy function are given as follows:
W ¼ 0; W ¼ 0; ð8Þ
In order to satisfy the rotationally-restrained boundary conditions, the approximate deflection function is uniquely constructed by linearly adding the associated weight coefficients xm (or en ) and 1 xm (or 1 en ) to the vibration characteristic functions of the simply-supported and clamped beams, respectively, as 1 X 1 X RR C mn X RR m ðnÞY n ðgÞ
@ F 2
@n
ðM nn Þn¼0 ¼ D11
ð10Þ ð11Þ
gÞ ¼ e
CC n ÞY n ð
gÞ þ ð1 e
gÞ
CC in which, X SS m ðnÞ and X m ðnÞ are the vibration eigenfunctions of simply-supported and clamped beams, respectively, along the x direction:
X SS m ðnÞ ¼ sin mpn CC X CC m ðnÞ ¼ sin am n sinh am n cm ðcos am n cosh am nÞ
cCC m ¼ ðsin am sinh am Þ=ðcos am cosh am Þ; am ¼ 4:7300; 7:8532; . . . Y SS n ð
ð12Þ
Y SS n ðyÞ ¼ sin n Y CC n ðyÞ ¼ sin n CC n ¼ ðsin n
ð13Þ
W 0 ðn; gÞ ¼ I0 X 0 ðnÞY 0 ðgÞ ð14Þ
Y 0 ðgÞ ¼ sinðspgÞ in which, I0 is the amplitude of imperfection, r and s are the numbers of half-wavelength in the longitudinal and transverse directions, respectively. The non-dimensional form of Airy function is given by
F ¼ N 1
g2 2
N2 k2
1 X 1 X n2 CC N12 kng þ vpq X CC p ðnÞY q ðgÞ 2 p¼1 q¼1
2 ðM gg Þg¼0 ¼ 4D22 ðM gg Þg¼1 ¼ 4D22
The initial imperfection is expressed as
X 0 ðnÞ ¼ sinðr pnÞ
" ðM nn Þn¼1 ¼ D11
2
pg a g sinh an g cCC n ðcos an g cosh an gÞ a sinh an Þ=ðcos an cosh an Þ;
ð19Þ
ð20Þ
ð21Þ
@2 W @n2 @2 W @n2
! þ kD16 n¼0
!
@2 W @ g2 @2 W @ g2
þ kD16 n¼1
!
g¼0
! g¼1
1 þ D26 k 1 þ D26 k
!
#
@W @n n¼0 n¼0 ! # 2 @ W @W ¼ ak1 @n@ g @n n¼1 n¼1 ! 3 @2W 5 ¼ bk1 @W @n@ g @ g g¼0 g¼0 ! 3 @2 W 5 ¼ bk2 @W @n@ g @ g g¼1
@2W @n@ g
¼ ak1
ð22Þ
ð23Þ
ð24Þ
ð25Þ
g¼1
For the laminates with only the rotational restraint spring at the boundary edges, it can be assumed that there is no variation about the deflection w ðWÞ with respect to y (g) and x (n) at the opposite longitudinal and transverse boundary edges, respectively. The boundary conditions in Eqs. (22)–(25) of the laminates against rotation at the four edges can be approximated by Eqs. (18)–(21). To make the displacement function Eq. (9) satisfying the boundRR ary conditions represented by Eqs. (18)–(21), W ¼ X RR m ðnÞY n ðgÞ is assumed and substituted into Eqs. (18)–(21). The associated weight coefficients xm can be determined from Eq. (18) or (19), and the associated weight coefficients en can be obtained according to Eq. (20) or (21). The weight coefficients are computed and given as follows:
ð15Þ
CC In the above equation, X CC p ðnÞ and Y q ðgÞ are the vibration eigenfunctions of clamped beam along the x and y directions, respectively. The dimensionless N 1 ; N 2 and N 12 are the applied longitudinal com-
ð18Þ
where, Mnn and Mgg are the non-dimensional form of the moment of Mxx and Myy , respectively. Since D16 – 0 and D26 – 0 are for the symmetrically-laminated anisotropic plates, the boundary conditions of moments at the rotationally-restrained boundary edges can be given by the following equations: "
and gÞ and gÞ are the vibration characteristic functions of simply-supported and clamped beams, respectively, along the y direction:
¼ ak1
g¼1
ðM nn Þn¼0 ¼ D11
Y CC n ð
c an ¼ 4:7300; 7:8532; . . .
ðM gg Þg¼1 ¼ D22
!
@W @n n¼0 @n2 n¼0 ! @2W @W ¼ ak 1 @n n¼1 @n2 n¼1 ! 2 @ W @W ¼ bk 2 @ g2 @ g g¼0 g¼0 ! @2W @W ¼ bk 2 @ g2 @ g g¼1
@2W
ðM gg Þg¼0 ¼ D22
Y RR n ð
ð17Þ
It can be seen that the assumed displacement function and Airy function satisfy the boundary conditions of Eqs. (16) and (17). In addition, the boundary conditions of moment at the longitudinal and transverse opposite edges of the symmetrically-laminated or orthotropic plates with odd half-wavelength in both the directions are expressed:
ðM nn Þn¼1 ¼ D11
SS CC X RR m ðnÞ ¼ xm X m ðnÞ þ ð1 xm ÞX m ðnÞ
@ F ¼ kN12 at g ¼ 0; 1 @n@ g
¼ k2 N2 ;
ð9Þ
where,
ð16Þ
2
m¼1 n¼1
SS nYn ð
@2F ¼ kN12 at n ¼ 0; 1 @n@ g
2
2.2. Deflection and Airy functions
Wðn; gÞ ¼
@2F ¼ N1 ; @ g2
xm ¼
en ¼
2D11 a2m ðsin am sinh am Þ ak1 mpðcos am cosh am Þ þ 2D11 a2m ðsin am sinh am Þ
2D22 a2n ðsin an sinh an Þ bk2 npðcos an cosh an Þ þ 2D22 a2n ðsin an sinh an Þ
ð26Þ ð27Þ
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Finally, the weight coefficients xm and en expressed in Eqs. (26) and (27) are substituted into the displacement function of Eq. (9). It is noted that it will result in two opposite edges simply supported (xm ¼ 1 or en ¼ 1) and two opposite edges clamped (xm ¼ 0 or en ¼ 0) when the rotational spring stiffness k1 or k2 equals to zero and infinite, respectively. 2.3. Solution By employing the Galerkin’s method in Eq. (7), the governing equation can be obtained:
Z
Z
1 0
1 0
" @4W @n4
4
þ a1
@ W @n3 @ g
4
þ a2
@ W @n2 @ g2
4
þ a3
4
@ W @ W þ a4 @n@ g3 @ g4
ð28Þ
By substituting the deflection function Eq. (9) and the Airy function Eq. (15) into the above equations and integrating from 0 to 1 with respect to n and g, the resulting governing equations in the matrix form can be obtained:
Vpq CL U
U
T
Kpq CNL U
Vpq CNL0 U
ð29Þ
¼0
in the above equation, m = 1, 2, . . ., M; n = 1, 2, . . ., N; p = 1, 2, . . ., P; q = 1, 2, . . ., Q; and
U ¼ C 11 C 12 . . . C ðm1ÞNþn . . . C MN1 C MN
h iT J ¼ w111 w112 w1mn w1MN1 w1MN w211 w212 w2pq w2PQ 1 w2PQ
mn T mn T Vmn w1mn ¼ Vmn Kmn EL U ENL þ KENL ENL0 V ELp
pq T pq T w2pq ¼ Vpq Kpq ð34Þ CL U CNL þ KCNL V CNL0 :
The convergence criterion of the nonlinear system is that the ratio of the magnitude of the vector DUn to the magnitude of the vector Unþ1 is smaller than the tolerance error.
ð35Þ
3. Numerical examples
þ b5
mn mn T mn mn Vmn ¼0 EL U U KENL U VENL0 U VELp U con
ð33Þ
the Jacobian matrix of the nonlinear system J can be expressed as
jDUn j 6 error: jUnþ1 j
@2F @2W @2F @2W @2F @2W @2F @2W 0 2 þ þ a5 2 2 @ g @n @n@ g @n@ g @n2 @ g2 @ g2 @n2 !# @2F @2W 0 @2F @2W 0 dWdndg ¼ 0 þ 2 @n@ g @n@ g @n2 @ g2 " Z 1Z 1 4 @ F @4F @4F @4F @4F þ b þ b þ b þ b 1 2 3 4 @n@ g3 @ g4 @n4 @n3 @ g @n2 @ g2 0 0 @2W @2W @2W @2W @2W @2W 0 @2W 0 @2W þ þ @n2 @ g2 @n@ g @n@ g @n2 @ g2 @n2 @ g2 !# @2W @2W 0 dFdndg ¼ 0 2 @n@ g @n@ g
d g1mn T T mn mn T mn T ¼ Vmn ðKmn EL ENL þ KENL ÞU VENL0 V ELp dU d g2pq T pq T pq pq T ¼ Vpq CL ðKCNL þ KCNL ÞU V CNL0 ; dU
v11 v12 . . . vðm1ÞNþn . . . vMN1 vMN
T
ð30Þ
The aforementioned nonlinear static and post-buckling simulation method is programmed in MATLAB. If no otherwise specified, the considered laminates in this study are constructed of the carbon/epoxy T800-3900-2 and the material properties of the lamina are given in [21] as: the longitudinal modulus EL ¼ 155:8 GPa, transverse modulus ET ¼ 8:89 GPa, in-plane shear modulus GLT ¼ 5:14 GPa, and Poisson’s ratio m ¼ 0:3. The width of the laminates is 0.5 m, and the ply thickness of the lamina is 0.125 mm. xx ; N yy and The nondimensional normal force coefficients N xy in the following are given by shear force coefficient N
2 xx ; N yy ; N xy ¼ Nxx ; Nyy ; Nxy b N 3 E2 h
ð36Þ
The non-dimensional longitudinal and transverse rotational spring stiffness K 1 and K 2 are used in the present study for the nonlinear static and post-buckling behavior of composite plates, and they are presented as:
K1 ¼
k1 a ; D11
K2 ¼
k2 b D22
ð37Þ
Kpq CNL Þ are shown in Appendix with all the terms being calculated numerically. The set of nonlinear equations can be written in a single expression as:
For the sake of convenience, the dimensionless parameters K ¼ K 1 ¼ K 2 are introduced if the non-dimensional longitudinal rotational spring stiffness K 1 equals the non-dimensional transverse rotational spring stiffness K 2 . The relatively short laminates are considered, i.e., it is assumed that there is only one half-wavelength in both the directions in the post-buckling mode of the laminate. The initial imperfection function with r ¼ s = 1 is introduced as:
h iT f ¼ g111 g112 g1mn ... g1MN1 g1MN g211 g212 ... g2mn g2MN1 g2MN
W 0 ðn; gÞ ¼ I0 sinðpnÞ sinðpgÞ
mn
and the detail of the constant ðcon Þ, vectors mn mn mn pq pq pq mn (Vmn EL ; VENL ; VENL0 ; V ELp ; VCL ; VCNL and VCNL0 ) and matrices ðKENL and
¼ ½0ðMNþPQÞ1 mn
mn mn T mn mn g1 ¼ Vmn EL U U KENL U V ENL0 U V ELp U con pq pq pq T pq g2 ¼ VCL U U KCNL U VCNL0 U
ð31Þ
and it can be solved by the Newton–Raphson method. It is assumed that the laminate is gradually (step-by-step) loaded with the applied load from zero to the final one. First, set the initial guess solution U1 ¼ ½0ðMNþPQ Þ1 , and then obtain the successive approximations to the solution from
Unþ1 ¼ Un J1 f ¼ Un þ DUn
ð38Þ
ð32Þ
with DUn ¼ Unþ1 Un ¼ J1 f, where J is the Jacobian matrix of the nonlinear system. Because the derivative of g1mn and g2pq in Eq. (30) with respect to the vector U are given as follows
The dimensionless central deflection W of the laminates is the maximum deflection, and it should contain the central imperfection, i.e.,
W ¼ Wð0:5; 0:5Þ þ W 0 ð0:5; 0:5Þ:
ð39Þ
3.1. Convergence and validity studies The results of the present method are compared with those obtained by employing the finite element method (FEM) in this section. The finite element analysis is performed by applying the commercial software ABAQUS; the element size of 0:0125 0:0125 m is used in FEM, and the shell element S4R is considered. First, the restrained square symmetrically-laminated orthotropic plates (RSSLOP) [0°/90°/0°/90°/0°/0°/0°/90°]S with all the edges
Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
Fig. 2. Convergence study for the load–deflection curves of RSSLOP with K ¼ 10 under pure in-plane shear.
rotationally restrained under pure in-plane shear loading are studied. The load–deflection curves with various approximation terms (i.e., M ¼ N ¼ P ¼ Q ¼ 3, 4, 5 and 6) for the shear post-buckling behavior of the RSSLOP with the dimensionless rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 10 are shown in Fig. 2, and the results are compared with those from FEM. As shown in Fig. 2, the results based on the approximation terms of 4–6 are almost identical, and the number of terms M ¼ N ¼ P ¼ Q ¼ 5 is thus used in the following nonlinear static and post-buckling analysis for RSSLOP. It is noted that the amplitude of the imperfection I0 ¼ 0:01 is selected in the analysis. In order to determine the effect of the amplitude of the imperfection I0 on the in-plane shear nonlinear static and post-buckling behavior of the RSSLOP with the restrained rotational stiffness K ¼ 10, different amplitudes of the imperfection are investigated in Fig. 3. It can be seen that the amplitude of imperfection I0 only has a relatively large influence on the early stage of the load–deflection curves. It can be expected that there is only a slight difference of the load–deflection patterns of laminates with very small amplitude of imperfection I0 and no imperfection. Thus, the amplitude of the imperfection I0 ¼ 0:01 is then selected in the following study in view of the stability in the convergence and accuracy of the results.
Fig. 3. Effect of the amplitude of imperfection I0 on the post-buckling behavior of RSSLOP with K ¼ 10 under pure in-plane shear.
121
Fig. 4. Post-buckling behavior of RSSLOP with various K values.
Fig. 5. Post-buckling behavior of RSSLOP with various K 1 values and constant K 2 ¼ 10.
The nonlinear static and post-buckling behavior of RSSLOP with various rotationally-restrained coefficient K values is examined in Fig. 4; while the nonlinear static and post-buckling behavior of RSSLOP with various dimensionless longitudinal rotational spring stiffness K 1 values and constant dimensionless transverse rotational spring stiffness K 2 ¼ 10 is evaluated in Fig. 5. It can be seen that the results agree quite well with those obtained from FEM, and the laminate will become stiffer gradually with the increase in the dimensionless rotational spring stiffness K (Fig. 4) or dimensionless longitudinal rotational spring stiffness K 1 (Fig. 5). The restrained square symmetrically-laminated anisotropic plates (RSSLAP) [45°/45°/45°/45°/45°/45°/0°/0°]S with all the boundary edges restrained against rotation and with imperfection amplitude of I0 ¼ 0:01 are considered. The convergence study of the nonlinear static and post-buckling response of RSSLAP with the dimensionless rotational spring stiffness of K ¼ K 1 ¼ K 2 ¼ 10 and 60 is given in Figs. 6 and 7, respectively. As shown in Figs. 6 and 7, there is a little discrepancy between the load-defection curves based on the approximation terms of M ¼ N ¼ P ¼ Q ¼ 4, 5 and 6 from Figs. 6 and 7, which is consistent with the results given in Fig. 2. Thus, the number of terms M ¼ N ¼ P ¼ Q ¼ 5 is chosen with confidence for the nonlinear static and post-buckling analysis in the following study. The load–deflection curves of RSSLAP with
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(a) Negative shear
(a) Negative shear
(b) Positive shear
(b) Positive shear
Fig. 6. Convergence study for the load deflection curves of RSSLAP with K ¼ 10 and I0 ¼ 0:01 under pure in-plane shear.
Fig. 7. Convergence study for the load deflection curves of RSSLAP with K ¼ 60 and I0 ¼ 0:01 under pure in-plane shear.
the rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 0; 10; 20; 40; 60; 1 under the positive and negative in-plane shear loading are shown in Fig. 8, and it can be seen that the present method can be used to investigate the nonlinear static and post-buckling behavior of the symmetric laminates, although the large rotational spring stiffness K (i.e., K ¼ 40; 60; 1) results in some relative discrepancies in the large deformation regime between the present analytical solution and FEM. In order to illustrate the computational efficiency and cost of applying the present semi-analytical method, the required computing time of the nonlinear static and post-buckling problem of symmetric laminates is shown in Table 1. The results are obtained using the workstation (Intel Xeon X5675 3.07 GHz processor, 64 GB). Because the computing time mainly relates to the approximation terms of displacement function and Airy function and the chosen number of the approximation terms M ¼ N ¼ P ¼ Q ¼ 5 is considered in this study for the nonlinear static and post-buckling analysis of both the symmetricallylaminated orthotropic and anisotropic plates as shown in the convergence study, the computational time (CPU) for the restrained square symmetrically laminated orthotropic plate (RSSLOP) [0°/ 90°/0°/90°/0°/0°/0°/90°]S with dimensionless rotational spring
stiffness K ¼ K 1 ¼ K 2 ¼ 10 are compared in Table 1 between the present semi-analytical method and FEM. As shown in Table 1, the results demonstrate that the computing efficiency of the present semi-analytical approach is dramatically improved when compared to the analysis using FEM. 3.2. Parametric study of the imperfect laminates under pure in-plane shear The effect of rotational restraint stiffness on the nonlinear static and post-buckling response of the orthotropic and symmetric laminates is examined in the above section, and the rotational spring stiffness have a major impact on the post-buckling stiffness of the laminates; i.e., it is easier to produce a large deflection for the laminates with smaller rotational restraint stiffness, and a small increment of shear is required for a given increment of central deflection for the laminates edged with a small rotational restraint stiffness. A laminate is a complex material that is often made of multiple layers of a particular lamina with different fiber orientations. Through investigating the nonlinear static and post-buckling behavior of the single lamina, the full understanding of the nonlinear static
Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
(a) Negative shear
123
Fig. 9. Effect of the orthotropy parameter a ¼ EL =ET on the post-buckling behavior of RSLSOP under pure in-plane shear.
(b) Positive shear Fig. 8. Validity study for post-buckling of RSSLAP with various K values and I0 ¼ 0:01 under pure in-plane shear.
Fig. 10. Effect of in-plane shear modulus GLT on the post-buckling behavior of RSLSOP under pure in-plane shear.
Table 1 CPU time for post-buckling analysis of RSSLOP.
CPU time (s)
Present semi-analytical method
FEM
5.50
80.50
response of the whole laminate can be manifested. First, the effect of material properties of the lamina on the nonlinear static and postbuckling behavior of single layer orthotropic plates is investigated. Then, the effect of fiber orientation on the nonlinear static and post-buckling behavior of single layer anisotropic plates is studied. It is noted that the thickness of the considered single layer square plates with a ¼ b = 0.5 is set as h = 0.002 m. The results presented in Fig. 9 show the effect of the orthotropy parameter a ¼ EL =ET on the nonlinear static and post-buckling behavior of the restrained single layer square orthotropic plates (RSLSOP) with the rotational spring stiffness K ¼ K 1 ¼ K 2 ¼ 10 and imperfection of I0 ¼ 0:01 under pure shear. The material properties of RSLSOP are: ET ¼ 8:89 GPa, GLT ¼ 5:14 GPa and mLT ¼ 0:3. As shown in Fig. 9, there is almost no difference on their post-buckling stiffness for RSLSOP with various orthotropy parameter a when under the pure in-plane shear action. The effect of the shear
Fig. 11. Effect of in-plane Poisson’s ratio RSLSOP under pure shear.
mLT on the post-buckling behavior of
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
modulus GLT on the nonlinear static and post-buckling response of the RSLSOP with the parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET and mLT ¼ 0:3 under pure in-plane shear is shown in Fig. 10. As shown in Fig. 10, the post-buckling stiffness of the orthotropic plate increases with the increasing shear modulus. While for the examples shown in Fig. 11, with the fixed parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa; EL ¼ 6ET and GLT ¼ 0:5ET , the effect of the in-plane Poisson’s ratio of RSLSOP is illustrated. As shown in Fig. 11, the variation of mLT does not make much influence on the post-buckling stiffness of RSLSOP. The effect of fiber orientation angle h on the nonlinear static and post-buckling behavior of restrained single layer square anisotropic plates (RSLSAP) [h] with the parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET ; GLT ¼ 0:5ET and mLT ¼ 0:3 under pure in-plane shear is examined (see Fig. 12). As shown in Fig. 12, the smaller the fiber orientation angle h, the larger the differences of the nonlinear static and post-buckling behavior between the applied positive and negative shear. The effect of the fiber orientation angle h on the nonlinear static and post-buckling response of the restrained square symmetrically laminated anisotropic plate (RSSLAP) ½h= h4S with parameters of K ¼ K 1 ¼ K 2 ¼ 10; I0 ¼ 0:01; ET ¼ 8:89 GPa, EL ¼ 6ET ; GLT ¼ 0:5ET and mLT ¼ 0:3 under pure in-plane shear is shown in Fig. 13. As depicted in Fig. 13, the nonlinear static behavior approaches to the one of orthotropic laminates when the number of layers (4 m) of RSSLAP ½h= hmS increases. As observed in Fig. 12, when the fiber orientation angle h increases (from the minimum h ¼ 0 up to the maximum of h ¼ 45 ), the postbuckling stiffness of RSLSAP under the positive in-plane shear will also increase; whereas the fiber orientation angle h shows less of an effect on the post-buckling stiffness of the RSLSAP under the negative in-plane shear. As expected, the post-buckling stiffness of RSSLAP ½h= h4S is proportional to the fiber orientation angle h (up to h ¼ 45 ) when the laminates are subjected to the positive or negative in-plane shear, which can also be obtained from Fig. 13.
Fig. 13. Effect of fiber orientation h on the post-buckling behavior of RSSLAP ½h= h4S under pure in-plane shear.
3.3. Parametric study for laminates under combined shear and compression The effect of loading ratio on the nonlinear static and postbuckling behavior of restrained square symmetrically laminated anisotropic plate (RSSLAP) ½45= 454S with the rotational restraint stiffness K ¼ 10 under combined shear and uniaxial compression or combined shear and biaxial compression loading is studied in Figs. 14 and 15, respectively. As shown in Figs. 14 and 15, the
Fig. 12. Effect of the fiber orientation angle h on the post-buckling behavior of RSLSAP ½h under pure in-plane shear.
Fig. 14. Effect of loading ratio on the post-buckling behavior of RSSLAP ½45= 454S under combined shear and uniaxial compression.
Fig. 15. Effect of loading ratio on the post-buckling behavior of RSSLAP ½45= 454S under combined shear and biaxial compression.
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
post-buckling stiffness of RSSLAP decreases with the increasing of the shear to compression loading ratios. 4. Conclusions In this study, the nonlinear governing equations for the nonlinear static and post-buckling behavior of imperfect symmetric laminates with all the edges elastically restrained against rotation and under pure in-plane shear and combined shear and compression are derived using the Galerkin method, and the Newton–Raphson method is employed to solve the nonlinear problem. The displacement functions are uniquely chosen by linearly combining the vibration eigenfunctions of simply supported and clamped beams to meet the rotationally-restrained boundary conditions in either the x- or y-direction of plate. As shown in validity study, the calculated results compare well with those obtained from FEM for both the orthotropic and symmetric laminates; while there are some small discrepancies between the load–deflection curves of the present method and FEM when the dimensionless rotational restraint stiffness becomes larger. A parametric study on the effect of a wide range of parameters to the nonlinear static and post-buckling behavior of restrained laminated plates is conducted, and the following conclusions are reached from the parametric study: (1) When under in-plane shear loading, the post-buckling stiffness of laminates increases with the shear modulus GLT ; while the orthotropic parameter EL =ET and Poisson’s ratio mLT do not much affect the post-buckling stiffness of laminates. (2) The fiber orientation angle h has a significant influence on the post-buckling stiffness of laminates under positive inplane shear, and the post-buckling stiffness of the symmetric laminates ½h= h4S under positive or negative in-plane shear is proportional to the fiber orientation angle h (up to h ¼ 45 ) when the fiber orientation angle increases from the minimum h ¼ 0 up to the maximum of h ¼ 45 . (3) The post-buckling stiffness of the laminates decreases with the increasing ratio of the shear to compression loading (for both the combined shear-uniaxial compression and combined shear-biaxial compression) when the plate is under the combined loading. In summary, a semi-analytical solution for the nonlinear static and post-buckling behavior of rotationally-restrained laminated plates with imperfection based on the Galerkin’s method is obtained, and it can efficiently and accurately perform the nonlinear static and post-buckling analysis as demonstrated by the comparisons with the results from the finite element analysis. Combined with the discrete plate analysis technique, the present solution can be potentially applied to nonlinear static and postbuckling analysis of FRP structural shapes, as similarly shown in a recent study using the numerical spline finite strip method by the authors [7]. Acknowledgment This research is partially supported by the National Natural Science Foundation of China (No. 51478265), and their financial support is gratefully acknowledged. Appendix A mn mn The detail of the constant ðconmn Þ, vectors (Vmn EL ; VENL ; VENL0 ; pq pq pq mn pq VCL ; VCNL and VCNL0 Þ and matrices (KENL and KCNL ) in governing equations are:
Vmn ELp ;
mn Vmn V2EL EL ¼ V1EL h i mn mn V1EL ¼ dði1ÞNþj ; V2EL ¼ ½01PQ 1MN Z 1 Z 1 " 4 RR @Y RR @ X i ðnÞ RR @ 3 X RR mn j ðgÞ i ðnÞ dði1ÞNþj ¼ Y ð g Þ þ a 1 j 4 3 g @ @n @n 0 0
@ 2 Y RR @ 3 Y RR @ 2 X RR @X RR j ðgÞ j ðgÞ i ðnÞ i ðnÞ þ a 3 2 2 @n g g3 @ @ @n # 4 RR @ Y j ðgÞ RR X m ðnÞY RR þ a4 X RR i ðnÞ n ðgÞdndg @ g4 þ a2
" Kmn ENL
¼
K11ENL
K12ENL
K21mn ENL
K22ENL
ðA:1Þ
# ;
K11ENL ¼ ½0MNMN ; K12ENL ¼ ½0MNPQ ; K22ENL ¼ ½0PQPQ ; h i mn K21mn ENL ¼ g ðp1ÞQ þq;ði1ÞNþj PQ MN Z 1 Z 1" 2 RR @ 2 Y CC q ðgÞ @ X i ðnÞ RR CC g mn ¼ a X ðnÞ Y j ðgÞ 5 ðp1ÞQþq;ði1ÞNþj p 2 @ g @n2 0 0 CC RR RR @X CC p ðnÞ @Y q ðgÞ @X i ðnÞ @Y j ðgÞ @n @n @g @g # 2 CC 2 RR @ X p ðnÞ CC @ Y j ðgÞ RR RR þ Y ð g ÞX ðnÞ X m ðnÞY RR q i n ðgÞdndg ðA:2Þ @ g2 @n2
2
mn Vmn V2ENL0 ENL0 ¼ V1ENL0 h i mn ; V2ENL0 ¼ ½01PQ V1mn ENL0 ¼ lði1ÞNþj 1MN Z 1 Z 1" 2 @ 2 Y CC mn j ðgÞ @ X 0 ðnÞ lði1ÞNþj ¼ a5 I0 X CC Y 0 ðgÞ i ðnÞ 2 @g @n2 0 0 @Y CC @X CC j ðgÞ @X 0 ðnÞ @Y 0 ðgÞ i ðnÞ @n @n @g @g # 2 CC 2 @ X i ðnÞ CC @ Y 0 ðgÞ RR þ Y ð g ÞX ðnÞ X m ðnÞY RR 0 j n ðgÞdndg @ g2 @n2 2
ðA:3Þ
mn V2ELp Vmn ELp ¼ V1ELp h i mn ; V2EL ¼ ½01PQ V1mn ELp ¼ eði1ÞNþj 1MN Z 1 Z 1" @Y RR @ 2 X RR @X RR j ðgÞ i ðnÞ RR i ðnÞ emn N1 Y ð g Þ þ 2kN 12 j ði1ÞNþj ¼ a5 2 @n g @ @n 0 0 # 2 RR @ Y j ðgÞ RR þ k2 N2 X RR ðA:4Þ X m ðnÞY RR i ðnÞ n ðgÞdndg @ g2 Z
1
Z
1
"
@ 2 X 0 ðnÞ
@X 0 ðnÞ @Y 0 ðgÞ Y 0 ðgÞ þ 2kN12 @n @g @n2 # 2 @ Y 0 ðgÞ RR ðA:5Þ X m ðnÞY RR þ k2 N2 X 0 ðnÞ n ðgÞ dndg @ g2
conmn ¼ a5 I0
0
Vpq CL ¼ V1CL
N1
0
V2pq CL
h i pq V2pq CL ¼ qði1ÞNþj 1PQ Z 1 " 4 CC @Y CC @ X i ðnÞ CC @ 3 X CC j ðgÞ i ðnÞ Y ð g Þ þ b 1 j 4 3 g @ @n @n 0
V1CL ¼ ½01MN ; qpq ði1ÞNþj ¼
Z
1
0
@ 2 Y CC @ 3 Y CC @ 2 X CC @X CC j ðgÞ j ðgÞ i ðnÞ i ðnÞ þ b 3 2 2 @n g g3 @ @ @n # 4 CC @ Y j ðgÞ CC X p ðnÞY CC þ b4 X CC i ðnÞ q ðgÞdndg @ g4
þ b2
ðA:6Þ
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Q. Chen, P. Qiao / Composite Structures 125 (2015) 117–126
" Kpq CNL ¼
K11pq CNL
K12CNL
K21CNL
K22CNL
# ;
K12ENL ¼ ½0MNPQ ; K21ENL ¼ ½0PQMN ; K22ENL ¼ ½0PQPQ ; h i pq K11pq ENL ¼ hðm1ÞNþn;ði1ÞNþj MNMN Z 1 Z 1 " RR RR @Y RR @X m ðnÞ @Y RR pq j ðgÞ n ðgÞ @X i ðnÞ hðm1ÞNþn;ði1ÞNþj ¼ b5 @n @ @n g g @ 0 0 # 2 RR @ 2 Y RR CC n ðgÞ @ X i ðnÞ RR Y j ðgÞ X CC ðA:7Þ X RR m ðnÞ p ðnÞY q ðgÞdndg 2 @g @n2 Vpq CNL0 ¼ V1CNL0
V2pq CNL0
h i pq V2pq CNL0 ¼ sði1ÞNþj 1PQ Z 1Z 1" 2 RR @ Y ð g Þ @ 2 X 0 ðnÞ j ¼ b5 I 0 X RR Y 0 ðgÞ i ðnÞ @ g2 @n2 0 0
V1CNL0 ¼ ½01MN ; spq ði1ÞNþj
@Y RR @X RR j ðgÞ @X 0 ðnÞ @Y 0 ðgÞ i ðnÞ @n @n @g @g # 2 RR 2 @ X i ðnÞ RR @ Y 0 ðgÞ CC Y ð g ÞX ðnÞ X p ðnÞY CC 0 j q ðgÞdndg: @ g2 @n2
þ2
ðA:8Þ
References [1] Paik JK, Kim DK, Lee H, Shim YL. A method for analyzing elastic large deflection behavior of perfect and imperfect plates with partially rotation-restrained edges. J Offshore Mech Arct Eng 2012:134. [2] Vescovini R, Bisagni C. Single-mode solution for post-buckling analysis of composite panels with elastic restraints loaded in compression. Compos Part B Eng 2012;43:1258–74. [3] Rhodes J, Harvey JM. The post-buckling behaviour of thin flat plates in compression with the unloaded edges elastically restrained against rotation. J Mech Eng Sci 1971;13:82–91. [4] Bisagni C, Vescovini R. Analytical formulation for local buckling and postbuckling analysis of stiffened laminated panels. Thin-Walled Struct 2009;47:318–34.
[5] Stamatelos DG, Labeas GN, Tserpes KI. Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels. ThinWalled Struct 2011;49:422–30. [6] Khong PW, Ong JH. Post buckling of a plate with elastic restraints at two unloaded edges. Comput Struct 1995;56:933–8. [7] Qiao P, Chen Q. Post-local-buckling of fiber-reinforced plastic composite structural shapes using discrete plate analysis. Thin-Walled Struct 2014;84:68–77. [8] Mittelstedt C, Erdmann H, Schroder KU. Postbuckling of imperfect rectangular composite plates under inplane shear closed-form approximate solutions. Arch Appl Mech 2011;81:1409–26. [9] Matthews FL, Zhang Y. Postbuckling behavior of anisotropic laminated plates under pure shear and shear combined with compressive loading. AIAA J 1984;22:281–6. [10] Zhang Y, Matthews FL. Large deflection behavior of simply supported laminated panels under in-plane loading. J Appl Mech 1985;52:553–8. [11] Kosteletos S. Postbuckling response of laminated plates under shear load. Compos Struct 1992;20:137–45. [12] Romeo G, Frulla G. Nonlinear analysis of anisotropic plates with initial imperfections and various boundary conditions subjected to combined biaxial compression and shear loads. Int J Solids Struct 1994;31:763–83. [13] Stein M. Postbuckling of long orthotropic plates in combined shear and compression. AIAA J 1985;23:788–94. [14] Stein M. Postbuckling of long orthotropic plates under combined loading. AIAA J 1985;23:1267–72. [15] Wang S, Dawe DJ. Spline FSM postbuckling analysis of shear-deformable rectangular laminates. Thin-Walled Struct 1999;34:163–78. [16] Chen QY, Qiao PZ. Post-buckling analysis of composite plates under combined compression and shear loading using finite strip method. Finite Elem Anal Des 2014;83:33–42. [17] Quatmann M, Reimerdes H-G. Preliminary design of composite fuselage structures using analytical rapid sizing methods. CEAS Aeronaut J 2011;2:231–41. [18] Beerhorst M, Seibel M, Mittelstedt C. Fast analytical method describing the postbuckling behavior of long, symmetric, balanced laminated composite plates under biaxial compression and shear. Compos Struct 2012;94:2001–9. [19] Chia CY. Postbuckling of composite plates under inplane compressive and shear loading having edges elastically restrained against rotation. Composite structures, vol. 3. Springer; 1985. p. 278–89. [20] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton, Florida: CRC Press; 2003. [21] Barbero EJ. Introduction to composite materials design. Boca Raton, Florida: CRC Press; 2010. [22] Chia CY. Nonlinear analysis of plates. New York, NY: McGraw-Hill; 1980.