INTRODUCTION
Sandwich plates have been used widely in many branches of engineering such as aerospace, shipbuilding, construction and other industries where strength, stiffness and weight are important. Sandwich panels commonly consist of two thin skins of high strength and stiffness surrounding a relatively thick and lightweight core. Usually, a panel might be made with skins of isotropic materials such as steel sheets or anisotropic materials such as carbon or glass fiber and an epoxy matrix, attached to isotropic or anisotropic core such as honeycomb, balsa or expanded foam core. The faceplates in the sandwich plates provide the primary load carrying capability, while the core carries the transverse shear loading.
Buckling of sandwich plate is an important issue in designing many structural
systems. It is one of the main modes of failure of these structures when subjected
to different work load conditions. Sandwich plate may buckle in various modes
depending on the material properties of the face sheets as well as the core
and their relative stiffnesses. To use sandwich plate efficiently, it is necessary
to develop appropriate models capable of accurately predicting the buckling
behavior. This phenomenon has been investigated by many researchers over the
past years (Rao, 1985; Kim and Hong,
1988; Ko and Jackson, 1993; Aiello
and Ombres, 1997; Hadi and Matthews, 1998; Cetkovic
and Vuksanovic, 2009). Allen (1969) for example used
a threelayered model for the analysis of sandwich beams and plates. However,
the analysis was based on the firstorder shear deformation theory. Higherorder
shear deformation theories have been employed to predict the buckling load of
sandwich plates (Frostig, 1998; Kant
and Swaminathan, 2004; Dafedar et al., 2003;
Pandit et al., 2008). Frostig
(1998) obtained local and general buckling loads for sandwich panels consisting
of two faces and a soft orthotropic core. Kant and Swaminathan
(2004) presented a displacement based higherorder formulation based on
an Equivalent Single Layer (ESL) theory, which cannot accurately predict the
local buckling modes. Dafedar et al. (2003) presented
analytical formulation to predict general buckling as well as wrinkling of a
general multilayer, multicore sandwich plate having any arbitrary sequence
of stiff layers and cores. However, these higherorder models that involve additional
displacement fields are computationally expensive in the sense that the number
of unknown to be solved is high compared to that of the firstorder shear deformation
theory.
Stiffeners had been used widely in the composite laminated panels to increase
the buckling load, improve the strength/weight ratios and reduce costs of structures.
A great deal of attention has been focused on plates reinforced by stiffeners
to improve their buckling behavior. Since the analysis of laminated composite
stiffened panels is complex; many researchers used numerical methods such as
FEM to clearly understand the buckling behavior of composite panels and to develop
some guidelines and curves, which will be helpful for the designers (Kolakowski
and Kubiak, 2005; Kim, 1996; Kang
and Kim, 2005; Perry et al., 1997; Bisagni
and Lanzi, 2002; Nemeth, 1997; Pecce
and Cosenza, 2000; Iyengar and Chakraborty, 2004;
Alinia, 2005; Mallela and Upadhyay,
2006).
This study is mainly concerned with the buckling behavior of simply supported rectangular sandwich plate. The sandwich panels consist of orthotropic core and composite laminated plate skin reinforced with isotropic bladestiffeners in multi directions. The stiffeners were added at the upper faceplates, while the lower faceplates were left without any stiffeners. The core of the sandwich panel is treated as threedimensional body. The firstorder Shear Deformation Plate Theory (SDPT) is used to represent the behavior of the face sheets whilst the stiffeners is modeled as simple beams with flexural stiffness only against outofplane bending, the face sheets are subjected to two types of loading, uniaxial and biaxial, loading. Due to the complexity of the problem, the energy method as well as the finite element method are used in this study. The results of the two methods have been compared for verification of the presented models. This study was conducted at Civil Engineering Department, Hashemite University, Jordan. The duration of the project is June 2008 to June 2010.
BASIC EQUATIONS AND PROBLEM FORMULATION
The elastic buckling load of a perfect sandwich stiffened panel (length a,
width b and thickness h) is computed using RayleighRitz method with the coordinates
xy along the inplane directions and z along the thickness direction (Fig.
1). The throughthickness variations of the displacement at
point (x, y, z) in the two faceplates are expressed as a function of midplane
displacements u, v, w and the independent rotations φ_{x} and φ_{y}
of the normal in xz and yz planes, respectively. The following basic assumptions
are used in the analysis: (1) the three layers forming the sandwich plate are
perfectly bonded together, (2) each layer is of uniform thickness (3) the material
of each layer is linearly elastic and (4) the strains in the sandwich plate
are small.

Fig. 1: 
Coordinate system for sandwich plate with multi directional
blade stiffeners 
The assumed displacement field, which satisfy the boundary condition of a simply
supported plate, is given by the form of fourier series:
Since, the core is located between the two face sheets its depth depends on
the displacement pattern of the face sheets and on the differences of the deflections
between the upper and the lower face sheets. The effects of the existence of
the highorder geometrical nonlinearities of the core can be neglected (Frostig
et al., 2005), which implies that the core actually can remains in
a linear state of deformations in spite of the large deformations of the sandwich
panel.
where, the subnotations u, l and c are refer to upper face plate, lower face
plate and core of the sandwich plate respectively. The unknown coefficients
(q_{mn}, A_{mn}, B_{mn}, R_{mn}, C_{mn},
D_{mn}, F_{mn}, J_{mn}) representing generalized displacements
amplitudes. The independent rotations can be represented by the form of fourier
series as:
The unknown coefficients (S_{mn}, G_{mn}, P_{mn}, N_{mn}) representing generalized rotations amplitudes. The displacements at a general point in each of the three zones can be expressed separately as:
The strains at general point can be expressed in term of linear and nonlinear strains. The linear part of straindisplacement relations {ε}_{L} has been used to derive the face plate's lamina property matrices. On the other hand, the nonlinear straindisplacement relations {ε}_{NL} have been employed to derive the geometric property matrices of the face plate's lamina:
The linear part of straindisplacement relations {ε}_{L} can be expressed separately for the upper and lower faceplates as:
where, the (ε^{u})_{L} and (ε^{l})_{L}
are refer to the linear strain in the upper and lower faceplates respectively.
The second order strain (ε)_{NL} for the upper and lower faceplates
is expressed as:
Herein, the core is modeled as a three dimensional solid element assuming the
inplane displacements vary quadratically through its thickness whilst the outofplane
varies linearly through the thickness (Yuan and Dawe, 2004).
The full set of the six component of linear strain which are taken into account
in the core are defined as:
The stressstrain relationship at a general point in the upper and lower faceplates for orthotropic laminated omposite material is defined as:
where, the (σ^{u}) and (σ^{l}) are refer to the stresses
in the upper and lower faceplates respectively. For the orthotropic material
which represents the behavior of the core, stresses at a material point in local
rectangular Cartesian coordinate axes are defined as:
According to the principle of conservation of energy, the potential energy,
Π^{i}, of a typical ith layer enclosing a space volume, V, can
be expressed as:
where, U^{i} represents the strain energy stored in the sandwich panels and W^{i} indicates the work done by externally applied stresses σ^{i}_{xx} and σ^{i}_{yy} acting in the x and y directions, respectively. The strain energy due to the two faceplates and core of the sandwich panels is given as:
For the case of sandwich panels with bladeisotropic stiffeners, two modes of buckling are usually considered, the local buckling of the plate between the stiffeners and the overall buckling (primary buckling) of the platestiffener combination. Herein, the derivation of buckling load is concern with the primary buckling. The assumed displacement field for the stiffeners is given by the form of fourier series:

Fig. 2: 
Crosssection of an eccentric stiffener 
where, Z_{n} and K_{n} are the location of the stiffeners in
the x and y direction respectively. The bending strain energy due to the stiffeners
can be given as:
where, I_{e} ^{n} and I_{e} ^{m} are the effective
moment of inertias about the axis of bending for the stiffeners in the x and
y directions, respectively. z_{c} is the distance from the plate middle
plane to the centroidal axis (through the centre of area) of a section consisting
of the stiffener and an effective plate area of width b_{e}. In the
case of an eccentric stiffener, the stiffener will lift the axis of bending
above the middle plane (Fig. 2). The effective moment of inertia
mentioned in Eq. 13 is an approximation, whose accuracy will
depend on the assumed value of b_{e}. It is found that z_{c}
= 0 is an acceptable value for eccentric stiffeners in many practical cases
as well. In practical design work, a z_{c}value calculated with a b_{e}
of about b_{e} = 20t has been suggested (Brubak,
2005; Brubak et al., 2007). Herein, the stiffeners
have been modeled as simple beams with flexural stiffness only against outofplane
bending. This simplification implies that possible torsional and local buckling
of stiffeners cannot be predicted. This may not represent a serious limitation
in practical cases since the practical constructional stiffener specifications
in typical design rules generally impose constructional design prevent any local
buckling of the stiffeners. Thus, the simplified stiffener model seems like
a reasonable one (Brubak and Hellesland, 2007).
The potential energy of the externally applied stresses σ^{i}_{xx} and σ^{i}_{yy} acting in the x and y directions for the upper and lower faceplates is given as:
By substituting the expressions for strain energy and the work done in Eq. 14, the potential energy for the sandwich panels with blade isotropic stiffeners can be expressed as follows:
where, U_{u}, U_{c}, U_{l}, U_{s} are the strain
energy of the upper faceplate, core of the sandwich plate, lower faceplate and
stiffeners in the x and y directions respectively, while the W is the potential
energy of the external applied stresses σ^{i}_{xx} and
σ^{i}_{yy} acting in the x and y directions on the top
and bottom faceplates. Substituting Eq. 10, 12,
14 into Eq. 15 and differentiation with
respect to the coefficients (q_{mn}, A_{mn}, B_{mn},
R_{mn}, C_{mn}, D_{mn}, F_{mn}, J_{mn},
S_{mn}, G_{mn}, P_{mn}, N_{mn}) and by setting
the variation in the total potential energy equal to zero .
The critical buckling load for sandwich plate with multiblade isotropic stiffeners
subjected to uniaxial or biaxial loads can be found. On other words, by minimizing
the total energy Π, the governing equations can be derived as:
where, the K and K_{G} are the standard stiffness and geometric stiffness
matrices, respectively. λ is the critical buckling stress. δ is the
vector of generalized degrees of freedom associated with the displacement field
functions This governing equation was incorporated into MATHEMATICA software
in order to determine the critical buckling stresses of the sandwich bladestiffened
panels.
FINITE ELEMENT MODELING FOR BLADESTIFFENED SANDWICH PANELS
Modeling composite stiffened sandwich panel needs extra attention in defining
the properties of the sandwich plate components. This type of modeling is associated
with numerical difficulties that require a very experienced user with a large
background and experience with nonlinear FEA modeling. However, this type of
analysis is very expensive in term of computational time and memory needed.
In the present study, Eigenbuckling analysis is performed for the sandwich
bladestiffened panels using a finite element package ABAQUS. The FE model is
composed of mainly eight noded quadrilaterals, stress/displacement solid elements
with largestrain formulation (C3D8) for all stiffened sandwich panel components
(faceplates, core and stiffeners) as shown in Fig. 3. Each
node has three degrees of freedom (ABAQUS, 2004). Tied
constrains were applied to obtain a full bond between the faceplates, core and
stiffeners. Simply supported boundary conditions were applied at all edges of
the faceplates and core by restraining the motion in the z direction and allowing
for free rotations. The composite sandwich stiffened panels are divided into
sufficient number of elements to allow for free development of buckling modes
and displacements. Some trial runs were also carried out to study the convergence
of the results. For uniaxial loading, the compressive loads were applied in
the x direction, while for biaxial loadings, the loads were applied in the x
and y directions of the faceplates as shown in Fig. 1.

Fig. 3: 
FE model for stiffened sandwich panel 

Fig. 4: 
FE model for isotropic sandwich panel (a) crosssection of
the hatstiffened sandwich pane andl (b) general view (FE mesh) 
Table 1: 
Comparison of results between current study and results available
in the literature for critical transverse buckling loads (N_{y})_{cr}
(kN/m) for simply supported doublehat stiffened sandwich plates 

In order to verify and validate the FEM described above, three simply supported
rectangular sandwich plates of deferent aspect ratio were considered here. A
single central double hatstiffener was added along the length (a = 2000 mm)
of the sandwich panel in the xdirection as shown in Fig. 4a
and b. The panels width is b, with b/a = 1, 1.5 and 2 in turn.
Transverse uniform stresses (σ_{y}) were applied at the upper and
lower faceplates. The isotropic material properties adopted in this example
for the faceplates and stiffeners are defined by Young's modulus E = 12.5 GPa
and Poisson's ratio v = 0.25. While the isotropic material properties of the
core are defined by Young's modulus E = 0.22 and Poisson's ratio v = 0.3 (Heder,
1993; Yuan and Dawe, 2004). Table
1 shows a comparison between the current study and results available in
the literature for the three stiffened sandwich panels. From these results,
it can be observed that the present study and the values available in the literature
are in good agreement. In this comparison, it's notable that the results provided
by Heder FE model are higher than the results provided by Yuan
and Dawe (2004) and the current model. In the FEM provided by Heder
(1993), the faceplates and stiffeners are modeled using fournode shell
elements, while the core is modeled using eightnode solid elements. It's noted
that the assembly of shell and solid elements is not fully compatible. Also
the method of applying boundary conditions and loads is somewhat different in
details from that specified in the current study. On the other hand, the results
provide by Heder (1993) analytical approach is lower
than the current study results. This is mainly due to the considerable level
of assumptions and approximations in his simple analytical model. This explains
most of the marginal difference between the results provided by Yuan
and Dawe (2004) and the current study in one side and the results provided
by Heder (1993) on the other side.
NUMERICAL EXAMPLE
The study, here, has been focused on the buckling behavior of simply supported stiffened sandwich plates subjected to inplane compressive loads (uniaxial and biaxial loads). A number of applications have been described, with the aim of demonstrating the capability and versatility of the presented approach. Unfortunately there is a shortage of earlier solutions with which to compare numerical results. In order to overcome this problem, the finite element model described in the previous section will be used to provide solutions for comparison with those arising from the semianalytical model. The material properties for the faceplates and the core of the sandwich panels are given in Table 2. In this table E_{1}, E_{2}, E_{3} are the modulus of Elasticity, G_{12}, G_{13}, G_{23} are the Shear modulus corresponding to the planes 12 , 13 and 23, respectively and v_{12}, v_{13}, v_{23} are the corresponding Poisson ratios. The adopted elastic material properties for the isotropic stiffeners in each computation are Young's modulus E = 200 GPa and Poisson's ratio v = 0.3.
Example 1: Sandwich plate with orthotropic facesheets and orthotropic core:
A square symmetric sandwich plates have been analyzed for predicting the critical
buckling stresses: The plate dimension (axb) is 1000 mmx1000 mm. The thicknesses
of the upper and lower facesheets are t_{u} = t_{1} = 5 mm.
Table 2: 
Material properties of the sandwich plate 


Fig. 5: 
Nondimensional buckling stress for orthotropic sandwich panels
subjected to uniaxial (σ_{yy}/σ_{xx} = 0) and
biaxial (σ_{yy}/σ_{xx} = 1) stresses 

Fig. 6: 
FE model for orthotropic sandwich panel subjected to uniaxial
stresses (a) general view (FE mesh) (b) first mode shape 
The thickness of the orthotropic core (t_{c}) is varied from 10 mm up
to 80 mm. The compression stresses were applied in uniaxial and biaxial directions as
shown in Fig. 1. The general buckling stresses obtained from
Eq. 16 were compared with the FE model and shown in Fig.
5. It can be concluded that the results calculated using both techniques
(Energy approach and FE analysis) are found to be fairly in good agreement.
The buckling mode shapes obtained are similar in respect with the buckling mode
shapes available in the literatures as shown in Fig. 6a and
b.
Example 2: Square stiffened sandwich plate with orthotropic facesheets
and orthotropic core: The first illustrative example considered here is
a square symmetric bladestiffened sandwich plates: The plate dimension (axb)
is 1000x1000 mm. The thicknesses of the upper and lower facesheets are similar
to the previous example t_{u} = t_{1} = 5 mm. The thickness
of the orthotropic core (t_{c}) is varied from 20 mm up to 80 mm. One
stiffener in each direction was added at the top of the faceplates as shown
in Fig. 7ac. The height of the isotropic
stiffeners is (h_{s}) = 50 mm, while the width of the stiffeners is
(b_{s}) = 5 mm. The face sheets were subjected to compression uniaxial
and biaxial stresses (σ_{yy}/σ_{xx} = 0.0, 0.5, 1.0
and 2). The nondimensional critical buckling stresses obtained using the semianalytical
model (Eq. 16) for stiffened sandwich plates were compared
with the FE model and shown in Fig. 8. It can be clearly seen
from the figure that the results calculated using both techniques (energy approach
and FE) are found to be fairly in good agreement.
The second example concerns with a square sandwich plate (a = b = 1000 mm)
stiffened with five blade stiffeners in each direction as shown in Fig.
9ac. All dimensions of stiffeners and sandwich panels
used in this example are similar to the dimensions mentioned in the previous
example.

Fig. 7: 
FE model for orthotropic sandwich panel stiffened with one
equally spaced isotropic stiffener in each direction subjected to uniaxial
stresses (a) location of the stiffeners (b) general view (FE mesh) and (c)
first mode shape 

Fig. 8: 
Nondimensional buckling stress for orthotropic sandwich panels
stiffened with one equally spaced isotropic stiffener in each direction
subjected to uniaxial and biaxial stresses 

Fig. 9: 
FE model for orthotropic sandwich panel stiffened with five
equally spaced isotropic stiffeners in each direction subjected to uniaxial
stresses (a) location of the stiffeners (b) general view (FE mesh) and (c)
first mode shape 
Again the stiffeners have been added at the top of the upper faceplate. The
upper and lower face sheets were subjected to compression uniaxial and biaxial
stresses (σ_{yy}/σ_{xx} = 0.0, 0.5, 1.0 and 2). The
nondimensional buckling stresses obtained using the energy approach (Eq.
16) were compared with the FE model and shown in Fig. 10.
It can be clearly seen that the comparison reveals very good correlation between
the results of the finite element model and the semianalytical model.
Example 3: Rectangular stiffened sandwich plate with orthotropic facesheets
and orthotropic core: A rectangular symmetric bladestiffened sandwich plates
have been analyzed as an illustrative example for predicting the critical buckling
stresses. The plate dimension (axb) is 2000 mmx1000 mm.

Fig. 10: 
Nondimensional buckling stress for laminated sandwich panels
stiffened with five equally spaced isotropic stiffeners in each direction
subjected to uniaxial and biaxial stresses 

Fig. 11: 
Nondimensional buckling stress for orthotropic sandwich panels
subjected to uniaxial (σ_{yy}/σ_{xx} = 0) and
biaxial (σ_{yy}/σ_{xx} = 1) stresses 

Fig. 12: 
FE model for orthotropic sandwich panel stiffened with one
isotropic stiffener in each direction subjected to uniaxial stresses (a)
general view (FE mesh) and (b) first mode shape 
The thicknesses of the
top and bottom facesheets are similar to the previous examples (t_{u} = t_{1} = 5 mm). The thickness of the orthotropic core (t_{c}) is varied from 20 mm
up to 80 mm. Initially the sandwich plates have been analyzed without adding
any stiffeners to the upper faceplates. The results of the nondimensional buckling
stresses under a compression uniaxial and biaxial loads conditions are shown
in Fig. 11. One stiffener in each direction was added at
the top of upper faceplates as a second illustrative example to compute the
critical stresses for the uniaxial and biaxial cases (σ_{yy}/σ_{xx}
= 0.0, 0.5, 1.0 and 2). The location of first stiffener at a/2, while the location
of the second stiffener at b/2 as shown in Fig. 12a and b.
The height of the isotropic stiffeners is (h_{s}) = 50 mm, while the
width of the stiffeners is (b_{s}) = 5 mm. Figure 13
shows the Nondimensional buckling stresses for the sandwich plates obtained
using the energy approach (Eq. 16) and FE model. It can be
obviously seen that the results are found to be in good agreement.
For all numerical examples showed in this study, it can be clearly seen that
adding stiffeners to the sandwich plates will improve their behavior and increase
their buckling resistance.

Fig. 13: 
Nondimensional buckling stress for orthotropic sandwich panels
stiffened with one isotropic stiffener in each direction subjected to uniaxial
and biaxial stresses 
In fact, stiffeners are one of the most economical
ways used to increase composite structures buckling resistance capability (Brubak
and Hellesland, 2007). It can be also seen from the figures that as the thicknesses of the core increases,
the nondimensional buckling stresses decreases. This doesn’t mean that
the over all buckling stress decreases. In fact, as the thicknesses of the core
increases the buckling stresses increases. It is worth mentioning that the comparison
between the semianalytical model and FE model reveals some notable discrepancies,
mainly with the sandwich plate supported by multistiffeners. This is mainly
due to the effect of the torsional stiffness which is accounted for in the finite
element model, but not in the energy method. This explains most of the marginal
difference between the results for both uniaxial and biaxial loading. It is
also worth mentioning that, the present method is found to be more economic
than a nonlinear FEM analysis (ABAQUS) in term of computational time and memory
needed to compute the critical buckling stresses of the same problem on the
same computer. This clearly demonstrates that the present method (Energy method)
is comparatively very efficient computationally.
CONCLUSION
In the present study, an approximate, semianalytical model has been derived
for stability analysis of simply supported sandwich plates reinforced with multiblade
stiffeners. As there is a shortage in investigation on the buckling of stiffened
sandwich panels, 3D nonlinear finite element model was presented and employed
to provide solutions for comparison with those arising from the semianalytical
model. The comparison reveals very good correlation between the results of the
finite element model and the semianalytical model. Most of the marginal differences
between the results are due to the effect of the torsional stiffness of the
stiffeners which is accounted for in the finite element model, but not in the
semianalytical model. The presented model is very efficient in terms of computational
time and memory needed compared to fully nonlinear finite element analysis.
Reinforcing sandwich panels with stiffeners could be an efficient and economical
way to increase their buckling resistance capability. Many new results are presented,
which helps to have some understanding regarding the structural behavior under
different situations. Also, these new results could be useful for future research
and comparisons.