Engineering Structures 28 (2006) 1307–1318 www.elsevier.com/locate/engstruct Differential quadrature nonlinear analysis of skew composite plates based on FSDT P. Malekzadeh a , G. Karami b,∗ a Department of Mechanical Engineering, School of Engineering, Persian Gulf University, Bushehr 75168, Iran b Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, United States Received 25 August 2005; received in revised form 21 December 2005; accepted 30 December 2005 Available online 28 February 2006 Abstract A differential quadrature (DQ) nonlinear analysis of skew laminated composite plates is presented. The governing equations are based on first-order shear deformation theory (FSDT). The geometrical nonlinearity is modeled using Green’s strain and von Karman assumptions. DQ discretization rules in association with an exact coordinate transformation are simultaneously used to transform and discretize the equilibrium equations and the related boundary conditions. The effects of skew angle, thickness-to-length ratio, aspect ratio and also the impact due to different types of boundary conditions on the convergence and accuracy of the method are studied. The resulting solutions are compared to those from other numerical methods to show the accuracy of the method with less computational effort. Also, numerical solutions for the large deflection behavior of antisymmetric cross ply skew plates under different geometrical parameters and boundary conditions are presented. c 2006 Elsevier Ltd. All rights reserved. Keywords: Nonlinear; Skew laminated plates; Differential quadrature method 1. Introduction Compared to small deflection analysis, nonlinear analysis of skew plates is quite cumbersome due to the complexity of the governing equations and the boundary conditions. There is a very strong singularity at the obtuse vertex along with increasing skew angle which must be dealt with properly. Most research on skew plate bending problems has been limited to small deflections theory and for isotropic materials. The nonlinear bending behavior of isotropic skew plates has been investigated by Kennedy [1] using the perturbation technique. Alwar and Ramachandra Rao [2,3] applied the method of dynamic relaxation to study the nonlinear behavior of clamped isotropic and orthotropic thin skew plates. Buragohain and Patodi [4] used a finite difference scheme with triangular mesh to study the large deflection behavior of thin, isotropic skew plates with fully clamped and simply supported edges. Srinivasan and Ramachandran [5,6] investigated the large deflection behavior of clamped variable thickness and also ∗ Corresponding author. Tel.: +1 701 231 5859; fax: +1 701 231 8913. E-mail address: [email protected](G. Karami). orthotropic thin skew plates by using the Green’s functions. Srinivasan and Boby [7] analyzed the nonlinear static behavior of thin, isotropic rhombic plates with clamped and simply supported edges by using a high precision conforming triangular plate bending element. Ray et al. [8] studied the large deflection of isotropic simply supported thin rhombic plates by applying the Galerkin technique and also conducted some experiments to verify their results. Pica et al. [9] presented a finite element analysis of the geometrically nonlinear behavior of plates using a Mindlin formulation to show the performance of different types of elements for isotropic skew, circular and elliptical plates. Recently, Duan and Mahendran [10] developed a new hybrid/mixed shell element for large deformation analysis of isotropic skew plates using oblique coordinate systems based on FSDT. Differential quadrature was introduced for structural analysis by Bert et al. [12] in 1988, and since then it has been employed for the analysis of various structural elements [13–22]. However, most of the applications of the differential quadrature method (DQM) were concerned with problems with linear differential equations. There exist few applications of the DQM for nonlinear analysis of 0141-0296/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2005.12.013
Compared to small deflection analysis, nonlinear analyof skew plates is quite cumbersome due to the complexitthe governing equations and the boundary conditions. Tis a very strong singularity at the obtuse vertex along wincreasing skew angle which must bedealt with properly. Mostresearch on skew plate bending problems has been limto small deflections theory and for isotropic materials. Thenonlinear bending behavior of isotropic skew plates has binvestigatedby Kennedy [1] using the perturbation techniqueAlwar and Ramachandra Rao [2,3] applied the method ofdynamic relaxation to study the nonlinear behavior of clamisotropic and orthotropic thin skew plates. Buragohain aPatodi [4] used a finite difference scheme with triangulamesh to study the large deflection behavior of thin, isotroskew plates with fully clamped and simply supported edgSrinivasan and Ramachandran [5,6] investigated the largedeflection behavior of clamped variable thickness and al
orthotropic thin skew plates by using the Green’s functioSrinivasan and Boby [7] analyzed the nonlinear static behavioof thin, isotropic rhombic plates with clamped and simplsupported edges by using a high precision conformtriangular plate bending element. Ray et al. [8] studied the largedeflection of isotropic simply supported thin rhombic plaby applying the Galerkin technique and also conducted sexperiments to verify their results. Pica et al. [9] presented afinite element analysis of the geometrically nonlinear behavioof plates using a Mindlin formulation to show the performanof different types of elements for isotropic skew, circular aelliptical plates. Recently, Duan and Mahendran [10] developeda new hybrid/mixed shell element for large deformatioanalysis of isotropic skew plates using oblique coordinsystems based on FSDT.
Differential quadrature wasintroduced for structuraanalysis by Bert et al. [12] in 1988, and since then ihas been employed for the analysis of various structelements [13–22]. However, most of the applications othe differential quadrature method (DQM) were concernwith problems with linear differential equations. There exfew applications of the DQM for nonlinear analysis
1308 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
-
d
d
f
of
picpicd byic
olvetes
. Lifhear
Thereles.ltingcyrentew
ngth, anM
.
m-efor-ally
ded
Nomenclature
a plate dimension in theξ -direction;Aij component of extensional stiffness of laminate;
Aξ(r)i j weighting coefficient of ther th-order derivative
in theξ -direction;Aξ
i j , Aηi j weighting coefficients of the first-order deriva-
tive in theξ - andη-directions, respectively;Bξ
i j , Bηi j weighting coefficients of the second-order deriva
tive in theξ - andη-directions, respectively;Bij bending-extensional stiffness of laminate;b plate dimension in theη-direction;Dij bending stiffness of laminate;Eij Young’s modulus of lamina;Gij shear modulus of lamina;g residual forces;h total thickness of laminate;f load vector;fi load vector at load stepi ;Kr
T tangent stiffness matrix defined in Eq.(32);Mx x , Myy , Mxy bending moment abouty- andx-axis and
twisting moment, respectively;M x x , M yy non-dimensional bending moment abouty-
and x-axis (=Mx x a2/D11h, Myya2/D11h), re-spectively;
Nξ , Nη number of grid points in theξ - andη-directions;Nx x , Nyy in-plane normal force resultant in thex- andy-
directions;(Nx x )i j , (Nyy)i j discretized in-plane normal force resul-
tant in thex andy-directions;Nxy in-plane shear force resultant;(Nxy)i j discretizedin-plane shear force resultant;N x x , N yy non-dimensional in-plane normal force resul-
tant in the x- and y-directions (=Nx x a2/D11,Nyya2/D11), respectively;
nx , ny thex- andy-component of the unit normal vectorto an arbitrary edge of the plate;
q intensity of distributed transverse load;Q non-dimensional transverse load parameter
(=qa4/D11h4);Qx , Qy transverse shear force in thex andy planes;R residual forces;TRN total residual norm;U the degrees of freedom vector or the generalize
displacement vector;Ur
i the degrees of freedom vector or the generalizedisplacement vector at iterationr of the load stepi ;
u, v,w displacement component in thex , y andtransverse direction of a point on mid-plane ofplate, respectively;
(u, v,w) displacement component in thex , y andtransverse direction of an arbitrary point(x, y, z)of plate, respectively;
x, y, z the Cartesian coordinate variables;
Wc non-dimensional center deflection(=w(a/2, b/2)/h);
β fiber orientation angle;ε the residual to applied force ratio;εo convergence tolerance;εi j strain components;εm membrane strain vector;εb bending strain vector;ϕx , ϕy bending rotation about thex andy axes;γi j shear strain components;γ transverse shear strain vector;η oblique coordinate variable;θ skew angle;σ byy non-dimensional bending stress at the center o
skew plate(= σbyya2(1 − υ2)/Eh2);σmyy non-dimensional membrane stress at the center
skew plate(= σmyya2(1 − υ2)/Eh2);υi j Poisson’s ratio of laminate;υ Poisson’s ratio of isotropic material;ξ oblique coordinate variable.
rectangular and circular plates, which are limited to isotroand orthotropic plates. The behavior of thin, circular, isotroelastic plates with immovable edges has been examineStriz et al. [14]. The nonlinear analysis of thin orthotroprectangular plates was studied by Bert et al. [15]. Lin et al. [16]employed the generalized differential quadrature to sthe problem of large deformation of thin isotropic plaunder thermal loading. Chen et al. [17] investigated the largedeflection analysis of thin orthotropic rectangular platesand Chen [18] studied the nonlinear free vibration analysis oorthotropic rectangular plates based on the higher-order sdeformation theory of Reddy.
This paper explores the utility of the DQM for nonlinearanalysis of skew composite plates based on FSDT.nonlinear equilibrium equations and boundary conditions adiscretized in an oblique coordinate system using DQ ruA Newton–Raphson scheme is used to solve the resunonlinear algebraic equations. The convergence and accuraof the presented algorithm are demonstrated via diffeexamples. Numerical results will be obtained for several skangles, a wide range of loads, different thickness-to-leratios, aspect ratios and boundary conditions. This workextension to some previous works by the authors on the DQlinear analysis of plate problems [19–22], is focused on thegeometrically nonlinear behavior of skew composite plates
2. The skew plate governing equations
The formulations for the nonlinear analysis of skew lainated composite plates based on the first-order shear dmation plate theory (FSDT) are presented. The geometricnonlinear theory, which includesnonlinearterms in thestrains,is considered. A skew plate composed of perfectly bonorthotropic layers of lengtha, width b and total thicknesshis considered (seeFig. 1).
P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318 1309
nt
an,,ou
tte
rs
ing,
inatetion
Fig. 1. The geometry of the skew composite plates.
In the first-order shear deformation theory, the displacemefield of a laminated composite plate is expressed as [23],
u(x, y, z, t) = u(x, y, t) + zϕx(x, y, t),
v(x, y, z, t) = v(x, y, t) + zϕy(x, y, t),
w(x, y, z, t) = w(x, y, t)
(1)
where (u, v,w) are the displacement components ofarbitrary point (x , y, z) of theplate in the laminated skew plateand (u, v, w) are the displacement projections on the mid-planeand(ϕx , ϕy) denote the rotations of the transverse normal abthe y- andx-axes, respectively.
Using the von Karman assumption [23], the straincomponents at an arbitrary point of the plates can be wrias follows:
εb =
εbx x
εbyy
εbxy
= z
∂ϕx
∂x∂ϕy
∂y∂ϕx
∂y+ ∂ϕy
∂x
= zκ,
εm =
εmx x
εmyy
γ mxy
=
∂u
∂x+ 1
2
(∂w
∂x
)2
∂v
∂y+ 1
2
(∂w
∂y
)2
∂u
∂y+ ∂v
∂x+ ∂w
∂x
∂w
∂y
γ ={γxz
γyz
}=
ϕx + ∂w
∂x
ϕy + ∂w
∂y
(2)
whereεb, εm andγ are the bending, membrane and transveshear components of strain, respectively.
Based on theFSDT, the force and moment resultants of theplates can be expressed as
NMQ
= A B 0
BT D 00 0 A
εm
κ
γ
(3)
t
n
e
whereA, B, D and A are the membrane, membrane-bendbending and shear stiffness matrices of the plate,
A =A11 A12 A16
A12 A22 A26A16 A26 A66
, B =
B11 B12 B16
B12 B22 B26B16 B26 B66
,
D =D11 D12 D16
D12 D22 D26D16 D26 D66
,
A =[
A55 A45
A45 A44
]=[κ55A55 κ45A45κ45A45 κ44A44
]
and
N =
Nx x
Nyy
Nxy
, M =
Mx x
Myy
Mxy
, Q =
{Qx
Qy
},
κi j is the shear correction factor.
Using Eqs.(2) and (3), the equilibrium equations of anarbitrary shaped composite plate in the rectangular coordsystem and in terms of displacement and bending rotacomponents can be derived as follows [23]:
A11∂2u
∂x2+ 2A16
∂2u
∂x∂y
+ A66∂2u
∂y2 + A16∂2v
∂x2 + (A12 + A66)∂2v
∂x∂y+ A26
∂2v
∂y2
+(
A11∂w
∂x+ A16
∂w
∂y
)∂2w
∂x2
+[(A12 + A66)
∂w
∂y+ 2A16
∂w
∂x
]∂2w
∂x∂y
+(
A66∂w
∂x+ A26
∂w
∂y
)∂2w
∂y2
+ B11∂2ϕx
∂x2+ 2B16
∂2ϕx
∂x∂y+ B66
∂2ϕx
∂y2+ B16
∂2ϕy
∂x2
+ (B12 + B66)∂2ϕy
∂x∂y+ B26
∂2ϕy
∂y2= 0 (4)
A16∂2u
∂x2+ (A12 + A66)
∂2u
∂x∂y+ A26
∂2u
∂y2
+ A66∂2v
∂x2+ 2A26
∂2v
∂x∂y+ A22
∂2v
∂y2
+(
A16∂w
∂x+ A66
∂w
∂y
)∂2w
∂x2
+[(A12 + A66)
∂w
∂x+ 2A26
∂w
∂y
]∂2w
∂x∂y
+(
A26∂w
∂x+ A22
∂w
∂y
)∂2w
∂y2
+ B16∂2ϕx
∂x2+ (B12 + B66)
∂2ϕx
∂x∂y+ B26
∂2ϕx
∂y2
+ B66∂2ϕy
∂x2+ 2B26
∂2ϕy
∂x∂y+ B22
∂2ϕy
∂y2= 0 (5)
1310 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
ini
ear
e
canby
belarateicalnal
ainanyes:
nd
s ofDQets
A55∂ϕx
∂x+ A45
∂ϕx
∂y+ A45
∂ϕy
∂x+ A44
∂ϕy
∂y
+ A55∂2w
∂x2+ 2A45
∂2w
∂x∂y+ A44
∂2w
∂y2
+ Nx x∂2w
∂x2+ Nyy
∂2w
∂y2+ 2Nxy
∂2w
∂x∂y+ q = 0 (6)
B11∂2u
∂x2+ 2B16
∂2u
∂x∂y+ B66
∂2u
∂y2+ B16
∂2v
∂x2
+ (B12 + B66)∂2v
∂x∂y+ B26
∂2v
∂y2
+(
B11∂w
∂x+ B16
∂w
∂y
)∂2w
∂x2
+[(B12 + B66)
∂w
∂y+ 2B16
∂w
∂x
]∂2w
∂x∂y
+(
B66∂w
∂x+ B26
∂w
∂y
)∂2w
∂y2− A55
∂w
∂x− A45
∂w
∂y
+ D11∂2ϕx
∂x2 + 2D16∂2ϕx
∂x∂y+ D66
∂2ϕx
∂y2 + D16∂2ϕy
∂x2
+ (D12 + D66)∂2ϕy
∂x∂y+ D26
∂2ϕy
∂y2 − A55ϕx − A45ϕ
y = 0
(7)
B16∂2u
∂x2+ (B12 + B66)
∂2u
∂x∂y+ B26
∂2u
∂y2
+ B66∂2v
∂x2+ 2B26
∂2v
∂x∂y+ B22
∂2v
∂y2
+(
B16∂w
∂x+ B66
∂w
∂y
)∂2w
∂x2
+[(B12 + B66)
∂w
∂x+ 2B26
∂w
∂y
]∂2w
∂x∂y
+(
B26∂w
∂x+ B22
∂w
∂y
)∂2w
∂y2− A45
∂w
∂x− A44
∂w
∂y
+ D16∂2ϕx
∂x2 + (D12 + D66)∂2ϕx
∂x∂y
+ D26∂2ϕx
∂y2 + D66∂2ϕy
∂x2
+ 2D26∂2ϕy
∂x∂y+ D22
∂2ϕy
∂y2 − A45ϕx − A44ϕ
y = 0. (8)
Eqs. (4)–(6) represent the balance of in-plane forcesthe x- and y-directions and the transverse shear forcesthe z-direction, respectively. Eqs.(7) and (8) represent thebending equilibrium equation about thex- and y-directions,respectively. Due to the anisotropic nature of the plate, thesequations are coupled. Also, the nonlinear terms that appethe bending equilibrium equations are due to anisotropy.
If the normal and tangent to an arbitrary edge of the skplate are denoted byn ands, respectively (shown inFig. 1), theboundary conditions along thisedge can be classified as [23]
eitherw = 0 or Qn = 0 (9)
n
in
w
eitherun = 0 or Nnn = 0 (10)
eitherus = 0 or Nns = 0 (11)
eitherϕn = 0 or Mnn = 0 (12)
eitherϕs = 0 or Mns = 0 (13)
where
un = nx u + nyv, us = −nyu + nxv,
ϕn = nxϕx + nyϕ
y, ϕs = −nyϕx + nxϕ
y
Qn = nx Qx + ny Qy,
Mnn = Mx x n2x + Myyn2
y + 2Mxynx ny,
Mns = nx ny(Myy − Mx x) + Mxy(n2x − n2
y).
The boundary condition at any edge of the platebe considered as a combination of conditions statedEqs.(9)–(13).
3. DQ analogs of the governing equations and boundaryconditions
The DQM requires the computational domain torectangular and it cannot be applied directly to irregudomains. To apply the DQM to such problems, a coordintransformation is necessary; that is, the irregular physdomain is transformed into a rectangular computatiodomain. For skew composite plateswith arbitrary laminate layup, the material points of skew plates in the physical domcan be transformed into the computational domain withoutapproximation, using the following linear transformation rul
x = ξ + (sinθ)η, y = (cosθ)η. (14)
Employing the chain rule for the spatial derivatives acoordinate transformation in(14), the derivatives in thephysical domain are expressed in terms of the derivativespace variables of computational domains. Simultaneously,discretization rules (seeAppendix A) can be used to discretizthe spatial derivatives in the computational domain. The resulfor spatial derivatives with different order are as follows:
∂( )
∂x
∣∣∣∣(ξi ,η j )
= ∂( )
∂ξ
∣∣∣∣(ξi ,η j )
=Nξ∑
m=1
Aξim( )mj ,
∂2( )
∂x2
∣∣∣∣(ξi ,η j )
= ∂2( )
∂ξ2
∣∣∣∣(ξi ,η j )
=Nξ∑
m=1
Bξim( )mj (15)
∂( )
∂y
∣∣∣∣(ξi ,η j )
=[− tanθ
∂( )
∂ξ+ secθ
∂( )
∂η
]∣∣∣∣(ξi ,η j )
= − tanθ
Nξ∑m=1
Aξim ( )mj + secθ
Nη∑n=1
Aηj n( )in (16)
∂2( )
∂x∂y
∣∣∣∣(ξi ,η j )
=[− tanθ
∂2( )
∂ξ2+ secθ
∂2( )
∂ξ∂η
]∣∣∣∣(ξi ,η j )
= − tanθ
Nξ∑m=1
Bξim ( )mj + secθ
Nξ∑m=1
Nη∑n=1
Aξim Aη
j n( )mn
(17)
P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318 1311
arb
as
∂2( )
∂y2
∣∣∣∣(ξi ,η j )
=[tanθ
∂2( )
∂ξ2− 2 secθ tanθ
∂2( )
∂ξ∂η
+ sec2 θ∂2( )
∂η2
]∣∣∣∣(ξi ,η j )
= tan2 θ
Nξ∑m=1
Bξim( )mj − 2 secθ tanθ
×Nξ∑
m=1
Nη∑n=1
Aξim Aη
j n( )mn + sec2 θ
Nη∑n=1
Bηj n( )in (18)
where the weighting coefficients are given inAppendix A.
The equilibrium equations as well as the related boundconditions can be discretized in the computational domainusing the DQ transformation rules(15)–(18). The DQanalogsof the equilibrium equations(4)–(8)become
where the in-plane resultant forces can be discretizedfollows:
(Nx x )i j =Nξ∑
r=1
Nη∑s=1
[(s41Aξir δ j s + s42δir Aη
j s)urs
+ (s43Aξir δ j s + s44δir Aη
j s)vrs + (s45Aξirδ j s
+ s46δir Aηj s)ϕ
xrs + (s47Aξ
irδ j s + s48δir Aηj s)ϕ
yrs]
+ A11
2
( Nξ∑r=1
Aξir wr j
)2
+ A12
2
×− tanθ
Nξ∑r=1
Aξirwr j + secθ
Nη∑r=1
Aηj rwir
2
+ A16
( Nξ∑r=1
Aξir wr j
)− tanθ
Nξ∑r=1
Aξirwr j
+ secθNη∑
r=1
Aηj rwir
,
(Nxy)i j =Nξ∑
r=1
Nη∑s=1
[(s51Aξir δ j s + s52δir Aη
j s)urs
+ (s53Aξir δ j s + s54δir Aη
j s)vrs + (s55Aξirδ j s
+ s56δir Aηj s)ϕ
xrs + (s57Aξ
irδ j s + s58δir Aηj s)ϕ
yrs]
+ A16
2
( Nξ∑r=1
Aξir wr j
)2
+ A26
2
×− tanθ
Nξ∑r=1
Aξirwr j + secθ
Nη∑r=1
Aηj rwir
2
+ A66
( Nξ∑r=1
Aξir wr j
)
×− tanθ
Nξ∑r=1
Aξirwr j + secθ
Nη∑r=1
Aηj rwir
,
(Nyy)i j =Nξ∑
r=1
Nη∑s=1
[(s61Aξir δ j s + s62δir Aη
j s)urs
+ (s63Aξir δ j s + s64δir Aη
j s)vrs + (s65Aξirδ j s
+ s66δir Aηj s)ϕ
xrs + (s67Aξ
irδ j s + s68δir Aηj s)ϕ
yrs]
1312 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
the
arylatesblebedary
siven
ary
t
ors
+ A12
2
( Nξ∑r=1
Aξirwr j
)2
+ A22
2
×− tanθ
Nξ∑r=1
Aξirwr j + secθ
Nη∑r=1
Aηj rwir
2
+ A26
( Nξ∑r=1
Aξirwr j
)
×− tanθ
Nξ∑r=1
Aξirwr j + secθ
Nη∑r=1
Aηj rwir
.
Eq.(7):Nξ∑
m=1
Nη∑n=1
(b11Bξ
imδ j n + b12Aξim Aη
j n + b13δim Bηj n)umn
+ (b21Bξimδ j n + b22Aξ
im Aηj n + b23δim Bη
j n)vmn
−s11Aξ
imδ j n + s12δim Aηj n −
Nξ∑r=1
Nη∑s=1
[(s45Aξir δ j s
+ s46δir Aηj s)Bξ
imδ j n + (b41Aξir δ j s + b42δir Aη
j s)
× Aξim Aη
j n + (s57Aξir δ j s + s58δir Aη
j s)δim Bηj n]wrs
wmn
+ (d11Bξimδ j n + d12Aξ
im Aηj n + d13δim Bη
j n)ϕxmn
+ (d21Bξimδ j n + d22Aξ
im Aηj n + d23δim Bη
j n)ϕymn
− A55ϕxi j − A45ϕ
yi j = 0. (22)
Eq.(8):Nξ∑
m=1
Nη∑n=1
(b21Bξ
imδ j n + b22Aξim Aη
j n + b23δim Bηj n)umn
+ (b31Bξimδ j n + b32Aξ
im Aηj n + b33δim Bη
j n)vmn
−s21Aξ
imδ j n + s22δim Aηj n
−Nξ∑
r=1
Nη∑s=1
[(s45Aξir δ j s + s46δir Aη
j s)Bξimδ j n
+ (b51Aξirδ j s + b52δir Aη
j s)Aξim Aη
j n
+ (s67Aξirδ j s + s68δir Aη
j s)δim Bηj n]wrs
wmn
+ (d21Bξimδ j n + d22Aξ
im Aηj n + d23δim Bη
j n)ϕxmn
+ (d31Bξimδ j n + d32Aξ
im Aηj n + d33δim Bη
j n)ϕymn
− A45ϕxi j − A44ϕ
yi j = 0. (23)
In the above equations the constant parametersai j , bi j , di j ,and si j , which depend both on the laminate stiffness andskew angle of the plate, are given inAppendix B; δi j representsthe Kronecker delta.
In a similar manner, the DQ analogs of the boundconditions can be obtained. In this paper, skew composite pwith immovable simply supported edge (S) and immovaclamped edge (C) and different combinations of them willconsidered. The physical conditions of these types of bounconditions are as follows.Simply supported edge:
w = 0, un = 0, us = 0, ϕs = 0, Mnn = 0. (24)
Clamped edge:
w = 0, un = 0, us = 0, ϕs = 0, ϕn = 0. (25)
For these types of edge conditions, the boundary conditionat the corners of the skew plate become similar to those gin Eq.(25).
The DQ analogs of the first four conditions in Eqs.(24)and(25)are as follows:
wi j = 0, (un)i j = nx ui j + nyvi j = 0,
(us)i j = −nyui j + nxvi j = 0,
ϕsi j = −nyϕ
xi j + nxϕ
yi j = 0 (26)
and those of the fifth conditions become
(Mnn)i j =Nξ∑
r=1
Nη∑s=1
[(c11Aξirδ j s + c12δir Aη
j s)urs
+ (c21Aξirδ j s + c22δir Aη
j s)vrs + (c31Aξir δ j s
+ c32δir Aηj s)ϕ
xrs + (c41Aξ
ir δ j s + c42δir Aηj s)ϕ
yrs]
+ e11
( Nξ∑r=1
Aξirwr j
)2
+ e12
− tanθ
Nξ∑r=1
Aξirwr j
+ secθNη∑
r=1
Aηj rwir
2
+ e13
( Nξ∑r=1
Aξir wr j
)− tanθ
Nξ∑r=1
Aξir wr j + secθ
Nη∑r=1
Aηj rwir
= 0 (27)
ϕni j = nxϕ
xi j + nyϕ
yi j = 0 (28)
where the expressions forci j and ei j are presented inAppendix B.
The discretized equilibrium equations and boundconditions can be written in the vector form as
R = g − f = 0. (29)
The components of the vectorsg andf are the left and righparts of Eqs.(19)–(23)and (26)–(28), respectively. To solvethe nonlinear equation(29) in a systematic manner, the vectof degrees of freedom or generalized displacement vector idefined as
U = {U} =[{u}T {v}T {w}T {ϕx}T {ϕy}T ]T (30)
P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318 1313
thnr taeft a
rie
m
id
dined
rafeuseat f
iththeor
tesricaling
d
own
ion.
n toareof
tsltant
endstheofsed
umtlysed
arendn
thees at
isad
esenthodod
od
thodpic
ws:
n
where
{u} =
u11u12...
uNξ Nη
, {v} =
v11v12...
vNξ Nη
, {w} =
w11w12...
wNξ Nη
,
{ϕx} =
ϕx11
ϕx12...
ϕxNξ Nη
, {ϕy} =
ϕy11
ϕy12...
ϕyNξ Nη
.
An incremental-iterative method should be used to solvenonlinear equation(29). In the present analysis, the solutioalgorithms are based on the Newton–Raphson method. Fopurpose, the load vectorf is applied incrementally, and forgiven value of the load, the Newton–Raphson iterations arbe continued until the required accuracy is reached. A briereview of the solution algorithm is as follows. Suppose thaiterationr for the load step fi+1, the vectorUr
i+1 gives theresidual forcesR(Ur
i+1) �= 0, then an improved value of Uis obtained by equating to zero the linearized Taylor’s seexpansion ofR(Ur+1
i+1 ) in the neighborhood ofUri+1 as
R(Ur+1i+1 ) = g(Ur
i+1) + KrT (Ur+1
i+1 − Uri+1) − fi+1 = 0 (31)
or
Ur+1i+1 = Ur
i+1 + (KrT )−1(fi+1 − g(Ur
i+1)) (32)
whereKrT is the tangent stiffness matrix evaluated atUr
i+1 andis given by
KrT = ∂g
∂U
∣∣∣∣U=Ur
i+1
=
∂g1
∂U1
∂g1
∂U2. . .
∂g1
∂UN∂g2
∂U1
∂g2
∂U2. . .
∂g2
∂UN...
......
...∂gN
∂U1
∂gN
∂U2· · · ∂gN
∂UN
∣∣∣∣∣∣∣∣∣∣∣∣∣U=Ur
i+1
.
N = 5Nξ × 5Nη is the total number of degrees of freedoand gi (i = 1, 2, . . . , N) are the components of vectorg. Ateach iteration the convergence is checked using a total resnorm criteria [9], i.e.,
ε = (RTR)1/2
(fTf )1/2 < εo. (33)
For each load step, the strain components, the benmoments and in-plane stresses resultants can be obtainusing Eqs.(2), (3), (15)and(16).
4. Numerical results
In this section, at first the convergence trends and accuof the presented algorithm are investigated through aexamples of skew plates with large enough loads to casignificant geometric nonlinearity. Since results for nonlinanalysis of skew composite plates are not available excep
e
his
to
t
s
ual
gby
cyweror
thin orthotropic skew plates, the results are compared wthose of isotropic and orthotropic skew plates to validateformulations and computer programs. Also, since exactclosed form solutions of nonlinear analysis of skew plaare not available yet, the results are compared with numeresults of other methods. At the end, the nonlinear bendbehaviors of antisymmetric laminated skew plates are presenteby analyzing different numerical examples.
The edges of the plate are numbered from 1 to 4 as shin Fig. 1. The first side is set atx = 0 and the other sidesare numbered consecutively in a counterclockwise directWithout loss of generality, thesame grid points along theξ -and η-axes are employed, namely,Nξ = Nη. In the entireproblems considered here, the individual layers are takebe of equal thickness and the shear correction factorstaken to be 5/6. Based on numeric experiment, any valuethe convergence toleranceεo ≤ 1 is sufficient to obtain anaccurate solution for the generalized displacement componenand resultant stresses. However, the accuracy of the resustresses can be slightly improved by using values ofεo � 1 andtherefore a value ofεo = 10−3 will be employed in all solvedexamples. The number of iterations for each load step depupon the number of grid points, the load step size and alsovalue of convergence tolerance. By increasing the numbergrid points, the number of iterations per load step is decreato a minimum value of three. In the examples, the maximnumber of iterations per load step is five. Unless explicimentioned, the following set of mechanical properties is uin the numerical studies:
E11
E22= 40,
G12
E22= 0.5,
G23
E22= 0.2, G13 = G12, υ12 = 0.25.
In all solved examples ten incremental load stepssufficient to obtain a converged solution; however, to fiaccurate intermediate solutions, in some cases more than teincremental load steps are used.
As a first example, the convergence and accuracy ofresults for the deflection, bending and membrane stressthe center of the skew isotropic plates are presented inTable 1.A clamped skewthin plate with relatively acute cornersconsidered and the results for two different values of loparameters are presented. Comparing the results of the prmethod with those of the lumped triangular element met(LTEM) [4], it is found that the converged results are in goagreement with those of the LTEM [4], and also nine gridpoints in each direction are sufficient to yield results with goaccuracy.
Toshow the convergence and accuracy of the present mefor composite skew plates, the nonlinear analysis of orthotroclamped skew plate considered by Alwar and Rao [2] isinvestigated here. The material properties used are as follo
E11
E22= 4.0,
G12
E22(1 − υ12υ21) = 0.35, υ12 = 0.3.
At first the convergence and accuracy of the center deflectiofor a skewplate with θ = 30◦ and a load parameterQ = 1,
1314 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
5335
87844
Table 1Convergence and accuracy of the stress components at the center of isotropic clamped skew plates (h/a = 0.001,b/a = 1, θ = 45◦, υ = 0.333)
for which the nonlinear effects are negligible, is consideredThe results are compared with those of Alwar and Rao2]and Iyengar and Srinivasan [11] in Table 2. It seems thatthe dynamic relaxation method (DRM) used by Alwar aRao [2] gives an upper bound for the deflections. Now consiskew plates with the same material properties as in previoexample, but with a large load parameter to cause a significimpact due to geometric nonlinearity. The results for tdeflection, resultant bending moments about thex- andy-axesand resultant membrane forces in thex- andy-directions at thecenter of the plates are presented inTable 3. This table alsoincludes the results obtained by the DRM [2].The results aregiven for two different skew angles and for different numbof grid points. Again one can see that the center deflectobtained by the DQM are slightly less than those of the DRbut still close agreement exists between the results of themethods for different parameters.
The convergence behaviors of the presented DQMnonlinear analysis of [0/90] antisymmetric cross ply skewcomposite plates subjected to a relatively large load parameteare considered as other examples. The results for thin clamantisymmetric cross-ply laminated skew plates are presentin Table 4. For skewplates with acute corners, i.e., the caθ = 60◦, the non-dimensional resultant bending moment abthe y-axis (M x x) has negligible value in comparison with thenon-dimensional resultant bending moment about thex-axis(M yy). It is found that in all cases nine grid points in eadirection yields accurate results for the center deflection aneleven grid points are sufficient to obtain accurate results fothe resultant forces and moments. The results for moderathick antisymmetric cross-ply laminated skew plates with tdifferent sets of boundary conditions, i.e., SSSS and SCare presented inTable 5. The fast rate of convergence of themethod is evident. It is found that in all cases eleven grid poin each direction give accurate results for the center deflectionand thirty grid points are sufficient to obtain accurate results fothe resultant forces and moments. The convergence behavior othe residual to applied force ratio, i.e.,ε (Eq. (33)), is shownin Table 6for different values of convergence toleranceεo andat three different load steps. Ten load steps are used.obvious that the number of iterations per load step dependthe convergence tolerance. The maximum number of iteratper load step is five. The non-dimensional central deflecand bending moment(M x x) for the convergence toleranc
r
t
s,o
r
ed
t
ly
,
s
isons
n
Table 2
Convergence of the center deflection(Wc × 103) for orthotropic clamped skewplates withQ = 1 (θ = 30◦, h/a = 0.001, b/a = 1)
Nξ = Nη [11] [2]7 9 13 19 25
0.4354 0.4285 0.4283 0.4283 0.4283 0.4283 0.43
Table 3Convergence of the results for orthotropic clamped skew plates withQ = 3200(h/a = 0.001,a/b = 1)
ε = 0.9, 0.01, 0.001 are 1.2119, 1.2105, 1.2105 and 18.0118.0012, 18.0025, respectively.
The DQM results will be compared with the solutionsa finite element method (FEM) large deformation analyof a skew plate here. The FEM solutions were prepausing ANSYS SHELL63 type elements. A clamped isotroskew plate with a skew angle of 60◦ is considered, forwhich the analytical solution can also be found [24]. The
P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318 1315
Table 5Convergence of the results for [0/90] antisymmetric laminated skew plates (Q = 1000,h/a = 0.1, b/a = 1)
Fig. 2. Comparisons between the DQM and different types of FEManalytical solutions for an isotopic clamped skew plate withθ = 60◦.
geometrical and material properties are those used by Dand Mahendran [10]; a/b = 1.5, h/a = 0.01, v = 0.333.In Fig. 2, this comparison is made by plotting the results fthe center deflection of the plate from the FEM and DQsolutions. In this figure also, the results from another hybFEM methodology by Duan and Mahendran [10], who haveused sixteen quadrilateral elements of equal size, arepresented. The DQM solutionshave a better agreement withose of the analytical solutions.
The effects of skew angle and boundary conditionsthe center deflection of the antisymmetric cross-ply thinmoderately thick skew plates are shown inFigs. 3–5. In eachcase it is obvious that the degree of hardening is increaseincreasing the skew angleθ . From these figures it is foun
d
an
d
lso
nd
by
Fig. 3. Non-dimensional center deflection of clamped [0/90] antisymmetriclaminated skew plates(a/b = 1).
that for the same value of center deflection simply supposkew plates need less load with respect to clamped plates. Aon increasing the load parameter, the geometric nonlineariteffects are increased significantly.
The influence of the number of layers (NL) and laminalay up on the center deflection of simply supported symmeand antisymmetric cross ply laminated skew plates is showFig. 6. The total thickness of the plate is constant. It is fouthat on increasing the number of layers, the stiffness of thantisymmetric cross ply skew plates increases; meanwhile, fthe symmetric cross ply skew plates the stiffness decreaseAlso, for the same number of layers, symmetric skew plaare stiffer than antisymmetric cross ply laminated plates.
The effects of aspect ratio on the non-dimensional cedeflection of CCSS [0/90] antisymmetric moderately thick
1316 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
]
l
t i
n of
elynte isare
hoseand
arythe
ofes
bleon
ndas
thate atarnon
sing
Fig. 4. Non-dimensional center deflection of simply supported [0/90antisymmetric laminated skew plates(a/b = 1).
Fig. 5. Non-dimensional center deflection of CSCS [0/90] antisymmetriclaminated skew plates(a/b = 1).
Fig. 6. The effects of number of layers and laminate lay up on non-dimensionacenter deflection of simply supported laminated skew plates (θ = 30◦, h/a =0.1, a/b = 1).
skew plates with relatively acute corners are shown inFig. 7.Thirty grid points are used in the numerical computations. Iobvious that on increasing the aspect ratio(b/a), the stiffnessof the plate is decreased.
s
Fig. 7. The effects of aspect ratio on the non-dimensional center deflectioCCSS [0/90] antisymmetric moderately thick skew plates (θ = 45◦, h/a =0.1).
5. Conclusions
The DQ nonlinear analysis of thin as well as moderatthick antisymmetric laminated skew plates with differeboundary conditions was investigated. Good convergencpresented even when only a small number of grid pointsused. The numerical results are in good agreements with tobtained by other numerical methods reported for isotropicorthotropic skew plates by other researchers.
For a given lay up of the lamina, the skew angle, boundconditions, aspect ratio and the thickness-to-length ratio ofplate greatly influence the deformation behaviors. The degreehardening increases with increase in skew angle, and decreaswith aspect ratio(b/a) and thickness-to-length ratio(h/a).
These studies indicate that the DQM is efficient and reliafor the nonlinear analysis of skew composite plates basedthe FSDT. Also, the utility of the DQM as an accurate aefficient method for solving complex nonlinear problems wbetter clarified.
Appendix A. DQ weighting coefficients
The basic idea of the differential quadrature method isthe derivative of a function, with respect to a space variabla given sampling point, is approximated as a weighted linesum of the samplingpoints in the domain of that variable. Iorder to illustrate the DQ approximation, consider a functif (ξ, η) having its field on a rectangular domain 0≤ ξ ≤ aand 0≤ η ≤ b. Let, in the given domain, the function valuebe known or desired on a grid of sampling points. Accordto the DQM, ther th derivativeof a function f (ξ, η) can beapproximated as
P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318 1317
nnthdgtin
b
a
From this equation one can deduce that the importacomponents of DQ approximations are weighting coefficieand the choice of sampling points. In order to determineweighting coefficients a set of test functions should be useEq.(A.1). For polynomial basis functions DQ, a set of Lagranpolynomials is employed as the test functions. The weighcoefficients for the first-order derivatives in thex-direction arethus determined as [13]
Aξi j = 1
a
M(ξi )
(ξi − ξ j )M(ξ j )for i �= j
−Nξ∑j=1i �= j
Aξi j for i = j
i, j = 1, 2 . . . , Nξ (A.2)
where
M(ξi ) =Nξ∏
j=1,i �= j
(ξi − ξ j ).
The weighting coefficients of second-order derivative canobtained as [13]
[Bξi j ] = [Aξ
i j ][Aξi j ] = [Aξ
i j ]2. (A.3)
In a similar manner, the weighting coefficients for they-direction can be obtained.
In the numerical computations, Chebyshev–Gauss–Lobquadrature points are used, that is [13]
1318 P. Malekzadeh, G. Karami / Engineering Structures 28 (2006) 1307–1318
.
pe:
;9
ith
ew
nit
ic
ly&
in1
;
r:
al
r
ch
er
trix
:
reeear
-hin
d
l
sis.
e11 = 1
2(n2
x B11 + n2y B12 + 2nxny B16),
e12 = 1
2(n2
x B12 + n2y B22 + 2nxny B26),
e13 = 1
2(n2
x B16 + n2y B26 + 2nxny B66). (B.6)
References
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