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Proceedings of the International Conference on Aerospace Science and Technology
26 - 28 June 2008, Bangalore, India
INCAST 2008- 075
BUCKLING BEHAVIOUR OF LAMINATED SANDWICH PLATES WITH SOFT
CORE BASED ON IMPROVED HIGHER-ORDER ZIG-ZAG THEORY
M. K. Pandit1, B. N. Singh2 and A. H. Sheikh3
1
Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur-721302, India,
m_pandit@naval.iitkgp.ernet.in
2
Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India,
bnsingh@aero.iitkgp.ernet.in
3
School of Civil, Environmental and Mining Engineering, The University of Adelaide, SA 5005, Australia,
ahsheikh@civeng.adelaide.edu.au
ABSTRACT: An improved higher order zigzag theory for buckling of sandwich plates is presented. The
present theory satisfies the conditions of transverse shear stress continuity at all the layer interfaces including
transverse shear stress free conditions at the top and bottom surfaces of the plate. The variation of in-plane
displacements through thickness direction is assumed to be cubic for both the face sheets and the core, while
transverse displacement is assumed to vary quadratically within the core but it remains constant over the face
sheets. The core is modeled as a three dimensional elastic continuum. An efficient C0 finite element is
proposed for the implementation of the improved plate theory. The accuracy and range of applicability of the
present formulation are established by comparing the present results with the standard results available in
literature.
1. INTRODUCTION
Sandwich plates are multilayered structures of low-density, flexible layers consisting of a core, laminated
and glued between high strength, stiff outer layers (facings). The facings provide the primary load carrying
capability, while the core carries the transverse shear loads. In order to have safe and reliable design, a clear
understanding of the behaviour and performance of these structures under different operating environments is
extremely important. One important feature of laminated sandwich is that it is very weak in shear due to its
low shear modulus compared to extensional rigidity. Also, it shows typical behaviour due to wide variation of
material properties between the core and face layers. Simple plate theories[4] developed based on the
appropriate representation of the shear deformation across the plate thickness, are not suitable for proper
modeling sandwich plates. In this regard, refined higher-order plate theories [1, 2] are quite accurate but their
performance is limited to the modeling of sandwich plates with transversely incompressible core where it is
assumed that the height of the core remains unchanged and the deflections at the upper and lower faces are
identical. However, modern sandwich plates/panels are usually made of compressible core like foam that are
associated with localized effect and considerable transverse normal deformation which aforementioned
theories lack to detect. Hence, to address all these effects properly an efficient plate theory should be used. In
this direction, there are a few efforts to take into account the effect of the transverse normal strain of the core
by different researchers [3, 6]. As the number of investigation considering all these aspects is not sufficient, the
objective of the present work is to develop an improved higher order plate model to study the buckling
behaviour of laminated sandwich plate with soft core. The variation of in-plane displacements is assumed to
be cubic with discontinuity in the transverse shear strain at the layer interfaces, which is extremely significant
in sandwich construction specifically at the interface between the core and face sheets. The transverse
displacement is assumed to vary quadratically within the core while it remains constant through the faces. The
core is considered to behave as a three dimensional elastic medium to incorporate the effect of transverse
normal deformation.
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Proceedings of the International Conference on Aerospace Science and Technology
26 - 28 June 2008, Bangalore, India
2. MATHEMATICAL FORMULATION
The through thickness variation of in-plane displacements may be expressed as follows
nu −1 nl −1
U = u + zθ x + ∑ ( z − ziu ) H ( z − ziu )α xu + ∑ ( z − z lj ) H (− z + z lj )α xl + β x z 2 + η x z 3
i j
(1)
i =1 j =1
nu −1 nl −1
V = v + zθ y + ∑ ( z − ziu ) H ( z − ziu )α yu + ∑ ( z − z lj ) H (− z + z lj )α yl + β y z 2 + η y z 3
i j
(2)
i =1 j =1
where, u, v denote the in-plane displacement of any point in the reference plane (plate mid-plane), θ x and θ y
are the rotations of the normal to the mid-plane about y-axis and x-axis respectively, nu and nl are number of
upper and lower layers respectively, β x , β y , η x , η y are the higher order unknowns, α xu , α xl , α yu , α yl are
i i i i
the change of slopes at the upper/lower i-th interface between i-th and (i+1)th layer, H ( z − ziu )
and H (− z + z lj ) are the unit step functions.
The transverse displacement is assumed to vary quadratically over the core thickness and constant over the
upper and lower face sheets and it may be expressed as
W = l1 wu + l2 w + l3 wl , for core
= wu , for upper faces (3)
= wl , for lower faces
where, wu , w and wl are the transverse displacement at the top, middle and bottom layers of the core,
respectively and l1, l2 and l3 are Lagrangian interpolation functions in thickness co-ordinate.
The stress-strain relationship of an orthotropic layer/lamina (say k-th layer) having any fibre orientation with
respect to structural axes system (x-y-z) may be expressed as
{σ } = ⎡Q k ⎤ {ε }
⎣ ⎦
where, {σ } , {ε } and ⎡Q k ⎤ are the stress vector, the strain vector and the transformed rigidity matrix of k-th
⎣ ⎦
lamina, respectively.
All the unknowns are expressed in terms of those at three planes (core mid plane, core face sheet interfacial
planes) by imposing transverse shear stress ( σ xz and σ yz ) continuity at all the interfaces including zero shear
at the plate top and bottom surface. The displacement field is then modified to make it suitable for C0 finite
element formulation. In this, the first order derivatives of wu and wl are assumed to be independent degrees of
freedom. Due to this assumption, the constraints are imposed which have been taken care of using a technique
based on penalty approach. Using strain-displacement relation and the modified displacement field, the strain
vector {ε } may be expressed as
T
⎡ ∂U ∂V ∂W ∂U ∂V ∂U ∂W ∂V ∂W ⎤
{ε } = ⎢ + + + ⎥ or, {ε } = [ H ]{ε } (4)
⎣ ∂x ∂y ∂z ∂y ∂x ∂z ∂x ∂z ∂y ⎦
where, {ε } is the mid-plane strain vector and the elements of [ H ] are functions of z and unit step functions.
A nine-noded quadrilateral C0 isoparametric element with 11 degrees of freedom (dofs) per node is used in
the present study. Using finite element approximation, the mid-plane strain vector {ε } may be expressed in
terms of {δ} containing nodal degrees of freedom for an element as
{ε } = [ B ]{δ } (5)
where, [B] is the strain-displacement matrix in the Cartesian co-ordinate system.
The elastic stiffness matrix of an element can be easily derived with the help of Principle of Virtual Work and
it may be expressed as
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Proceedings of the International Conference on Aerospace Science and Technology
26 - 28 June 2008, Bangalore, India
⎡K e ⎤ =⎡ke ⎤ + ⎡ k p ⎤ ,
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (6)
∫∫ [ B] [ D ][ B ] dxdy ,
T
where, ⎡ke ⎤ =
⎣ ⎦ (7)
n
[ D ] = ∑ ∫ [ H ]T ⎡Q k ⎤ [ H ]
⎣ ⎦
dz , (8)
k =1
and ⎡ k p ⎤ is the penalty matrix.
⎣ ⎦
Similarly, the geometric stiffness matrix [ K Ge ] of an element can be derived and it may be expressed as
[ K Ge ] = ∫∫ [ B ] [G ][ B ]dxdy
T
(9)
n
where, [G ] = ∑ ∫ [ H G ]T ⎡ S k ⎤ [ H G ] dz , the stress matrix [Sk] contains the in-plane stress components of the k-th
⎣ ⎦
k =1
layer and the matrix [ H G ] is similar to [ H ] .
After evaluating the element stiffness and geometric stiffness matrices for all the elements, they have been
assembled together to form the overall stiffness matrix [ K ] and geometric stiffness matrix [ K G ] of the plate.
The governing equation of buckling may be expressed as
[ K ]{φ} = λ [ KG ]{φ} (10)
where, λ is the critical buckling load and {φ } are global displacement vectors defining buckling mode shapes.
3. RESULTS AND DISCUSSION
In order to show the performance of the present improved plate model, examples of the laminated
sandwich plates subjected to applied in-plane compressive load have been analyzed and the results obtained
have been compared with the results available in the literature.
3.1 Rectangular sandwich plate with laminated face sheets
A simply supported rectangular sandwich plate with cross-ply laminated face sheets
(0/90/0/90/C/90/0/90/0) is considered for example. It is subjected to uniaxial compressive load ( N x ). The
plate is studied for three different aspect ratios, a/b=0.5, 1.0 and 2.0 using a number of mesh sizes to show the
convergence characteristics. The individual ply of the face sheet possesses same thickness and material
properties (E11=181GPa, E22=10.3GPa, G12=7.17GPa and ν12=0.28). The core (G13=0.146GPa,
G23=0.0904GPa with negligible in-plane rigidities) has a thickness of hc=10h/11. The through thickness direct
modulus of the core is taken to be E11=1.46GPa. The buckling load parameters, λ = Ν x b 2 Ε 22 hc3 obtained in
the present analysis for hc/b=0.01 are presented with those obtained by Rais-Rohani and Marcellier[5] in Table
1. The table clearly shows that the present results are in well agreement with the analytical solution[5] with
excellent convergence requiring reasonably less number of elements to get the desired accuracy.
Table 1 Buckling load parameters ( λ = Ν x b 2 Ε 22 hc3 ) of a simply supported rectangular sandwich plate
(0/90/0/90/C/90/0/90/0) with laminated cross-ply face sheets
Reference Aspect ratios (a/b)
0.5 1.0 2.0
Present (4×4)m 11.1773 5.8492 5.8507
Present (6×6) 11.1687 5.8446 5.8448
Present (8×8) 11.1673 5.8438 5.8439
Present (10×10) 11.1672 5.8436 5.8436
Present(12×12) 11.1670 5.8435 5.8435
Rais-Rohani and Marcellier [5] 11.086 5.830 5.830
m
: Quantities within the parenthesis indicates mesh division
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Proceedings of the International Conference on Aerospace Science and Technology
26 - 28 June 2008, Bangalore, India
3.2 Effect of different parameters on the buckling loads of a square sandwich plate
In this example, a square sandwich plate having a dimension of 225×225 mm2 and consisting of two
single layered orthotropic face sheets of thickness 0.2 mm each and a core of 10 mm thick, is considered. The
material properties of the core and face sheets are as follows: Core: Ec11=0.02 GPa, Ec22=0.02 GPa, Ec33 = 0.2
GPa, Gc12 = 0.0146 GPa, Gc13 = 0.146 GPa, Gc23 = 0.0904 GPa, ν12=0.3; Face sheets: E11 = 229 GPa, E22
=13.35 Gpa, G12=5.25 GPa and ν12=0.3151. Figs. 1 and 2 show the effect of different parameters on the
buckling load where the buckling load verses ply orientation angle is shown for different cases. The different
parameters considered are ply orientation, boundary condition and ratio of in-plane forces (Ny/Nx and Nxy/Nx).
The boundary conditions considered are SSSS (all edges simply supported) and CCCC (all edges clamped).
Ny/Nx=1.0, Nxy/Nx=0.5
400 Ny/Nx=1.0, Nxy/Nx=1.0
300 Ny/Nx=1.0, Nxy/Nx=0.5
Ny/Nx=1.0, Nxy/Nx=2.0
280 Ny/Nx=1.0, Nxy/Nx=1.0 360
Ny/Nx=1.0, Nxy/Nx=2.0
260
320
Buckling load, Ncr
240
Buckling load, Ncr
220 280
200
240
180
200
160
140 160
120
120
100
0 20 40 60 80 100 0 20 40 60 80 100
Fiber orientation, θ Fiber orientation, θ
Fig. 1 Variation of buckling load with fiber orientation for SSSS boundary condition Fig. 2 Variation of buckling load with fiber orientation for CCCC boundary condition
It is observed from the Figs. 1 and 2 that for both the CCCC and SSSS condition the buckling loads first
increase then again decrease when the fiber orientation angle increases for all the ratio of in-plane loadings.
Moreover, the buckling loads decrease when the ratio Nxy/Nx increases from 0.5 to 2.0 for constant ratio of
Ny/Nx for SSSS condition. Similar observation can also be made in case of CCCC boundary condition.
4. CONCLUSION
In the present investigation, an improved higher order zigzag plate theory is proposed for the accurate
prediction of the buckling load of soft core sandwich plate. The proposed zigzag plate theory possesses the
advantages of single layer and layer wise theories including the transverse compressibility of the core. A nine
noded isoparametric element with 11 dofs per node is employed to implement the proposed plate model. The
results obtained in the form of buckling loads of the sandwich plate structure show an excellent performance
of the present formulation with good convergence characteristics.
REFERENCES
[1] Cho M and Parmerter RR. Efficient higher order composite plate theory for general lamination configurations. AIAA
J. 1993, 31(7), 1299-1306.
[2] Di Scuiva M. A refined transverse shear deformation theory for multilayered anisotropic plates. Atti Academia
Scienze Torino, 1984, 118, 279-295.
[3] Frostig Y, Baruch M, Vinley O and Sheinman I. High-order theory for sandwich-beam behaviour with transversely
flexible core. J Engg Mech. 1992, 118(5), 1026-1043.
[4] Ghugal YM and Shimpi RP. A review of refined shear deformation theories of isotropic and anisotropic laminated
plates. J Reinf Plastics Compos. 2002, 21( 9), 779-813.
[5] Rais-Rohani M and Marcellier P. Buckling and vibration analysis of composite sandwich plates with elastic
rotational edge restraints. AIAA J. 1999, 37, 579-587.
[6] Yuan WX and Dawe DJ. Overall and local buckling of sandwich plates with laminated faceplates, part II:
Applications. Comput Methods Appl Mech Eng. 2001, 190, 5215–5231.
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