Jtam-A4.dvi
JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
56, 1, pp. 3-14, Warsaw 2018
DOI: 10.15632/jtam-pl.56.1.3
HYGROTHERMOELASTIC BUCKLING RESPONSE OF COMPOSITE
LAMINATES BY USING MODIFIED SHEAR DEFORMATION THEORY
Masoud Kazemi
Environmental Sciences Research Center, Islamshahr Branch, Islamic Azad University, Islamshahr, Iran
e-mail: masoud kazemi@hotmail.com; kazemii@iiau.ac.ir
In this study, a finite element based formulation is developed for analyzing the buckling
and post-buckling of composite laminates subjected to mechanical and hygrothermal loads
usingModifiedHyperbolic ShearDeformationTheory (MHSDT).The changes in the critical
buckling loadarepresented for different lamination schemes, thicknesses,material properties
andplate aspect ratios. In addition, post bucklinganalysis is performed for a compositeplate
subjected to uniform in-plane thermal and moisture induced loadings by using MHSDT.
Matlab softwarehasbeenused forprogrammingtheanalysis.The resultsobtainedbyMatlab
codes are in a satisfactory consistence compared to the references. Thus, the developed
MHSDT has been validated for buckling and post buckling analysis of laminated plates in
hygrothermal environment.
Keywords: angle-ply laminate, buckling, composite plate, finite element method, shear de-
formation theory
1. Introduction
Compared to conventional metal structures, fibrous composite materials continue to experience
increased application in aerospace,marine, automobile and othermechanical and civil structures
due to their superior strength and stiffness toweight ratios; however, due tomaterial anisotropy,
analyzing and designing these materials are more complicated thanmetallic materials.
In order to prevent buckling and post-buckling effects in laminated plates, using an extra-
-strength is of great practical importance in the structural design of laminated plates.
Buckling is known as one of themost critical failuremodes, often pre-generated or produced
during service life. A significant reduction inweight of laminated plates can be achieved conside-
ring the post bucklingbehavior,which is an important factor in aerospace structures.The elastic
buckling and post-buckling of fiber reinforced composite plates are investigated in several text-
books (Agarwal et al., 2006; Reddy, 2004; Turvey andMarshall, 2012). Composite laminates are
also susceptible to delamination buckling and exterior damage at stress free edges, which occurs
when the properties mismatch at the ply interface. It can also be produced by external forces,
elevated temperature and absorbedmoisture. Stresses within laminates are redistributed to re-
duce the load carrying capacity, when delamination occurs. Composite laminates are subjected
to changing environmental conditions like temperature andmoisture. The effect of temperature
and moisture is known as thermal and hygroscopic effect, respectively. The combined effect of
these two parameters is called the hygrothermal effect. A hygrothermal environment reduces
both strength and elastic properties, especially in the case of fibrous polymeric composites. Fur-
thermore, associated hygrothermal expansion, either alone or in combination withmechanically
produced deformation, can result in buckling, large deflections, and high stress levels. Conse-
quently, examining the hygrothermal effects is essential in analyzing and designing laminated
systems (Tauchert and Huang, 2012). Due to the fact that most applications are limited to
purely thermal loadings, the majority of published researches lie in this field.
4 M. Kazemi
According to the similarities between mathematical formulations of the governing ther-
mal and hygroscopic loadings, the given thermoelastic solutions could be generalized to elasto-
-hygrothermal cases. Similarly, it is not difficult to simplify the hygrothermal formulations and
solutionmethods to include the isothermal effects. Forpredicting the real behavior of a structure,
it is important to choose an adequate theorywhich is used in the expansion of different variables
(Mantari et al., 2012). In the 3D elasticity theory, heterogeneous laminated plates are modeled
as 3D solid elements, so predicting transverse shear stresses can be significantly improved, ho-
wever, by using this theory would lead to a complex procedure and multiplied computational
cost.
In the literature, differentmodels have been suggested for studying the composite laminated
structures, including layerwise, quasi-layerwise and equivalent single layer models. Three prin-
cipal equivalent theories have been proposed to reduce the 3D models to 2D ones; which are
knownas theClassical LaminatedPlateTheory (CLPT), First-order ShearDeformationTheory
(FSDT) and Higher-order Shear Deformation Plate Theory (HSDT) (Kharazi et al., 2014).
In the CLPT, which relies on the Love-Kirchhoff assumptions, the transverse shear defor-
mation is neglected and is only applicable for thin laminated plates, so, in order to consider
the shear effect, the FSDT based onReissner-Mindlin theory has been developed. The FSDT is
simple to perform and can be applied for both thick and thin laminates; however, the accuracy
of solutions strongly relies on the shear correction factors. In addition, the FSDTwould not give
satisfactory results in predicting the accurate and smooth variations of stresses, specifically for
laminated plates with clamped or free edges, sharp corners and highly skewed geometry where
high stress gradients occur. To overcome the limitation of the FSDT, a simple higher order the-
ory was presented by Reddy (2004) for laminated plates, various types of HSDT, which include
higher order terms in Taylor’s expansion.
Many studies in the literature investigated the buckling and post-buckling in composite la-
minated thin plates subjected to mechanical or thermal loadings or both based on the classical
plate theory, see for example (Kazemi and Verchery, 2016; Peković et al., 2015; Ahmadi and
Pourshahsavari, 2016; Muc and Chwał, 2016). In some other studies (Girish and Ramachan-
dra, 2005; Mechab et al., 2012; Dafedar and Desai, 2002), the application of shear deformation
plate theories was developed for buckling and post-buckling analysis of laminated plates under
combined mechanical and thermal loading. It should be noted that in all these investigations,
the material properties are considered to be independent of temperature. Although compre-
hensive literature has been published in the field of pure mechanical or pure thermal loadings,
few investigations have been devoted to the elastic buckling and post-buckling caused by co-
upled thermal and mechanical loads, which is encountered in real cases and operational life of
composite structures.
A refined two-dimensional model was proposed byBrischetto (2013) for static hygrothermal
analysis of laminated composites and sandwich shells neglecting the transverse shear deforma-
tion effects. Sreehari andMaiti (2015) introduced a finite element solution for handling buckling
and post buckling analysis of laminated plates under mechanical and hygrothermal loads using
a refinedHSDT; however, the accuracy of the method was verified only for cross-ply laminates.
Natarajan et al. (2014) considered the effect of moisture condensation and thermal variation on
the vibration and buckling of laminates with cutouts within the formulation of the extended
finite element method. Pandey et al. (2009) examined the influence of moisture concentration,
temperature variation, plate parameters andfiber-volume fraction on the buckling andpost buc-
kling of the laminated plates based onHSDT and vonKarman’s nonlinear kinematics; however,
the distribution of temperature andmoisture on the surface was assumed to be uniform.
The aim of present work is to analyze the buckling and post buckling behavior of composite
laminated plates in hygrothermal environment using the Finite Element Method (FEM) based
on a newhigher order formulation, in which the displacement of themiddle surface is developed
Hygrothermoelastic buckling response of composite laminates... 5
as a trigonometric and exponential function of thickness, and the transverse displacement is
assumed to be constant through the thickness. An appropriate distribution of the transverse
shear strain is assumed across the plate thickness and, also, the stress-free boundary conditions
are considered on the boundary surface, therefore, a shear modification factor is not needed.
2. Trigonometric shear displacement model (TSDM)
A laminated plate consisting of N orthotropic plies is considered. Length, width and thickness
of the rectangular plate are a, b, and h, respectively. An 8-noded serendipity quadrilateral
element, which is C0-continuous isoperimetric bi-quadratic, has been used for discretization of
the laminated plate. In this work, the following new displacement model is proposed to satisfy
the boundary conditions at the top and bottom of the laminated plate
u(x,y,z) =u0(x,y)−z
∂w
∂x
+
[
sin
πz
h
exp
(
mcos
πz
h
)
+
π
h
mz
]
θx(x,y)
v(x,y,z) = v0(x,y)−z
∂w
∂y
+
[
sin
πz
h
exp
(
mcos
πz
h
)
+
π
h
mz
]
θy(x,y)
w(x,y,z) =w0
(2.1)
where u, v, w represent displacement components in the x, y and z directions, respectively;
and u0, v0, w0 are displacement components in the middle surface of the plate. θx and θy are
rotations about the y and x axes at the mid-plane, respectively. The first order derivatives of
the transverse displacement can be formulated in terms of the in-plane displacement parameters
as separate independent degrees of freedom as given below
u(x,y,z) =u0(x,y)−zφx(x,y)+ [g(z)+Γz]θx(x,y)
v(x,y,z) = v0(x,y)−zφy(x,y)+ [g(z)+Γz]θy(x,y)
w(x,y,z) =w0(x,y)
(2.2)
where
φx =
∂w
∂x
φy =
∂w
∂y
g(z) = sin
πz
h
exp
(
mcos
πz
h
)
Γ =
π
h
m
The linear displacement vector given in the above equation can be expressed in terms of the
middle surface of the laminated plate as follows
ε5×1 =Z5×13ε13×1 (2.3)
where
ε=
{
ε01 ε
0
2 ε
0
6 κ
1
1 κ
1
2 κ
1
6 ε
0
4 ε
0
5 κ
2
4 κ
2
5
}T
ε01 =
∂u0
∂x
ε02 =
∂v0
∂y
ε06 =
∂u0
∂y
+
∂v0
∂x
ε04 =
∂w0
∂y
−φy
ε05 =
∂w0
∂x
−φx k
0
1 =Γ
∂θx
∂x
−
∂φx
∂x
k02 =Γ
∂θy
∂y
−
∂φy
∂y
k06 =Γ
(∂θx
∂y
+
∂θy
∂y
)
−
∂φx
∂y
−
∂φy
∂x
k11 =
∂θx
∂x
k12 =
∂θy
∂x
k16 =
∂θy
∂x
+
∂θx
∂y
k24 = θy k
2
5 = θx
6 M. Kazemi
Z=
1 0 0 z 0 0 g(z) 0 0 0 0 0 0
0 1 0 0 z 0 0 g(z) 0 0 0 0 0
0 0 1 0 0 z 0 0 0 g(z) 0 0 0
0 0 0 0 0 0 0 0 0 1 0 g(z) 0
0 0 0 0 0 0 0 0 0 0 1 0 g(z)
and
ε13×1 =L13×7∆7×1 ∆=
{
u0 v0 w0 θx θy φx φy
}T
The following assumptions are considered in the derivation of the equations:
• Small elastic deformations are assumed (i.e. deformations and rotations are small andagree
to the Hooke’s law).
• The plies of the composite laminated structure are supposed to be well bonded.
The linear strain equations derived from the displacements of Eqs. (2.1), which are valid for thin
as well as thick plates under consideration, are as follows
εxx = ε
0
xx+zε
1
xx+sin
πz
h
exp
(
mcos
πz
h
)
ε2xx
εyy = ε
0
yy +zε
1
yy +sin
πz
h
exp
(
mcos
πz
h
)
ε2yy
εxy = ε
0
xy +zε
1
xy +sin
πz
h
exp
(
mcos
πz
h
)
ε2xy
εxz = ε
0
xz +
π
h
(
cos
πz
h
−msin2
πz
h
)
exp
(
mcos
πz
h
)
ε3xz
εyz = ε
0
yz +
π
h
(
cos
πz
h
−msin2
πz
h
)
exp
(
mcos
πz
h
)
ε3yz
(2.4)
and
ε0xx =
∂u
∂x
ε1xx =m
π
h
∂θx
∂x
−
∂2w
∂x2
ε2xx =
∂θx
∂x
ε0yy =
∂v
∂x
ε1yy =m
π
h
∂θy
∂x
−
∂2w
∂x2
ε2yy =
∂θy
∂x
ε0xy =
∂v
∂x
+
∂u
∂y
ε1xy =m
π
h
∂θy
∂x
+m
π
h
∂θx
∂y
−2
∂2w
∂x∂y
ε2xy =
∂θy
∂x
+
∂θx
∂y
ε0xz =m
π
h
θx ε
3
xz = θx ε
0
yz =m
π
h
θy ε
3
yz = θy
(2.5)
3. Governing equations of the hygrothermal buckling and post-buckling
The laminated plate composed of elastic orthotropic plies and the stress–strain relations in the
orthotropic local frame are as follows (Reddy, 2004)
σ1
σ2
τ12
τ13
τ23
=
Q11 Q12 0 0 0
Q12 Q22 0 0 0
0 0 Q66 0 0
0 0 0 Q55 0
0 0 0 0 Q44
ε1
ε2
γ12
γ13
γ23
(3.1)
where Qij are elastic stiffness coefficients relative to the plane-stress state that neglects the
transversal stress. These coefficients are given below (Reddy, 2004) in terms of the engineering
constants in the material coordinates
Hygrothermoelastic buckling response of composite laminates... 7
Q11 =
E1
1−ν12ν21
Q22 =
E2
1−ν12ν21
Q12 = ν12Q11 Q33 =G12
Q44 =G23 Q55 =G13 ν21 = ν12
E2
E1
(3.2)
In general, the laminates are in the plane stress state due to temperature or moisture changes;
therefore, externally applied stresses would develop at the supports. These in-plane stresses can
be evaluated using elasto-hygrothermal constitutive equation. When hygrothermal effects are
considered, the stress tensor is usually expressed in the contracted notation as follows
σi =Qij
(
εj −
T
∫
T0
αj(τ,M) dτ −
M
∫
M0
βj(T,m) dm
)
i,j =1,2,3 (3.3)
where the elastic stiffness coefficientsQij, the thermal expansion coefficientsαj, and themoistu-
re coefficients βj depend upon the temperature T andmoisture concentrationM. For moderate
temperature∆T =T−T0 andmoisture∆M =M−M0 changes from the corresponding stress-
-free values T0 andM0, if the elastic properties are considered independent from the hygrother-
mal, the stress-strain relations are simplified as follows
σ=Qij(ε1−αj∆T −β∆M)(ε1−αj∆T −β∆M) i,j =1,2,3 (3.4)
Proper tensor transformations can be employed in transforming equation (3.4) from principal
material coordinates x1, x2 and x3 to the plate coordinates x, y and z. For a typical k-th ply of
the laminate, the resulted expression can be written as
σxx
σyy
τxy
τxz
τyz
k
=
Q11 Q12 Q16 0 0
Q12 Q22 Q26 0 0
Q16 Q26 Q66 0 0
0 0 0 Q55 Q54
0 0 0 Q45 Q44
k
εxx−αx∆T −βx∆C
εyy −αy∆T −βy∆C
εxy−αxy∆T −βxy∆C
εxz
εyz
k
(3.5)
or in a condensed form
σk =Qkεk (3.6)
whereQij, αi, βi (i,j =x,y,xy) denote the transformedmaterial coefficients.
According to the potential energy theorem, the equilibrium state can be achieved when
variation of the total potential energy equates to zero.
The potential energy theorem can be expressed for the typical i-th ply enclosing a space
volume V as follows
∫
V
(σxxδεxx+σyyδεyy + τxyδεxy + τxzδεxz + τyzδεyz) dVe−
∫
A
qδw dAe =0 (3.7)
When the laminate is subjected to temperature or moisture changes, due to the restriction on
freeing the hygrothermal loading, some stresses are developed at the supports. The governing
equations on the pre-buckling can be obtained via the following formula
K∆=F (3.8)
whereK is the linear stiffnessmatrix andF represents the load vector associated with the tem-
perature variation or hygroscopic effects. Equation (3.8) is solved under the specified boundary
condition and in-plane loads. In the next step, the geometric stiffness matrix KG associated
8 M. Kazemi
with these in-plane loads is calculated. The critical hygrothermal buckling is calculated through
solving the linear eigenvalue problem
(K+λcrKG)∆=0 (3.9)
The smallest eigenvalue corresponds to the amplitude of the critical buckling load. In the post-
-buckling step, the nonlinear stiffness matrixKnl is incorporated as
(K+Knl+λcrKG)∆=0 (3.10)
The geometric stiffness matrix can be expressed as
KG =σ
p
xKG1+σ
p
yKG2 (3.11)
where σpx, σ
p
y denote externally applied stresses acting in the x and y directions. Subsequently,
the critical buckling stresses can be calculated by the following formulas
σpxcr=λcrσ
p
x σ
p
yc
r=λcrσ
p
y (3.12)
4. Numerical results and discussion
In this Section, numerical examples are presented for bucklingandpostbucklingof the laminated
compositeplates undermechanical andhygrothermal loads.Theaccuracyof theproposedTSDM
model considering the transverse shear stresses is examined. A variety of problems are solved
using the finite element formulation and the results are compared with 3D elasticity solution. It
is important to note that the proposed displacement model can be applied to any lay-up of the
laminated plates. The different mechanical properties examined in the numerical examples are
given in Table 1.
Table 1.Material properties used in the numerical examples
Mater- Elastic constants
ial No. (Reddy and Liu, 1985; Dafedar and Desai, 2002)
1 E1/E2 =25,G12 =G13 =0.5E2,G23 =0.2E2, ν12 =0.25
2 E1/E2 =3 to 40,E3 =E2,G12/E2 =G13/E2 =0.60, G23/E2 =0.50,
ν12 = ν23 = ν13 =0.25
3 E1/E2 =40,E3 =E2,G12/E2 =G13/E2 =0.50, G23/E2 =0.20,
ν12 = ν23 = ν13 =0.25
4 E1/E2 =15,E3 =E2,G12/E2 =G13/E2 =0.50, G23/E2 =0.3356,
ν12 = ν23 = ν13 =0.3, a1/a0 =0.015, a2/a0 = a3/a0 =1.00
5 Elastic moduli of graphite/epoxy ply at different moisture
concentrations C [%],
E1 =130GPa, G13 =G12 =6.0GPa, G23 =0.5G12,
ν12 = ν23 = ν13 =0.3, β1 =0 and β2 =β3 =0.44 and
C [%] 0.00 0.25 0.50 0.75 1.00 1.25 1.50
E2 [GPa] 9.50 9.25 9.00 8.75 8.50 8.50 8.50
In order to simplify comparison, the critical buckling stresses have been transformed into
dimensionless coefficients as follows
λcr =
σcrb
2
E2h2
(4.1)
Hygrothermoelastic buckling response of composite laminates... 9
4.1. Examples for validating the TSDM model
Three cases are examined to confirmTSDM formulation using finite element programming.
Case A
Asymmetric four-layered (0/90/90/0) cross-ply laminated plate is considered under uniaxial
compression loading. The critical buckling coefficients for various values of length-to-thickness
ratios a/h are presented in Table 2. As it is demonstrated in Table 2, the HSDT overestimates
the critical buckling loads in comparisonwith the results from the present formulation and those
given by Pagano et al. (1994).
Table 2.Effect of length to thickness ratio on the critical buckling load
a/h Present
3D HSDT
(Pagano and Reddy, 1994) (Reddy and Liu, 1985)
5 1.922 1.575 1.997
10 13.367 13.453 13.384
20 20.689 21.707 21.886
50 23.354 23.356 23.747
100 24.034 24.255 24.953
Case B
The effect of elastic moduli ratios on the buckling loads of a square plate under uniaxial
loading is examined, and the results are presented in Table 3. According to the results obtained
via the TSDM formulation are in excellent agreement with other references.
Table 3.Effect of elastic moduli ratios on critical buckling loads
E1/E2 Present
3D HSDT
(Pagano and Reddy, 1994) (Reddy and Liu, 1985)
3 5.396 5.399 5.114
10 9.952 9.967 13.384
20 15.327 15.352 15.297
30 19.703 19.758 19.968
40 23.564 23.451 23.344
Case C
Table 4 presents the comparison between the critical buckling coefficients obtained through
the present model and the reference values for the square laminated plate under uniaxial com-
pression loading. The analysis is carried out for two values of fiber orientation angles θ = 30◦
and θ=45◦ for both of the two-ply and six-ply antisymmetric angle-ply laminates. The results
are validated by comparing themwith the HSDTmodel proposed by Reddy and Liu (1985).
4.2. Effect of the length-to-thickness ratio on the critical buckling load
Asymmetric four-layered (0/90)s cross-ply laminated plate is consideredunderbothuniaxial
and biaxial compression loadings. The effect of the side-to-thickness ratio for the simply suppor-
ted rectangular plate is examined usingmaterial No. 1, and the results are plotted in Fig.1. It is
10 M. Kazemi
Table 4.Critical buckling coefficients for angle-ply laminates
θ=30◦ θ=45◦
a/h
2 ply 6 ply
a/h
2 ply 6 ply
Present [10] Present [10] Present [10] Present [10]
5 10.694 11.543 13.404 13.536 5 10.084 10.782 12.169 12.172
10 16.108 17.123 29.046 33.624 10 16.734 18.051 30.648 32.504
20 18.234 18.764 41.023 46.231 20 19.234 19.764 48. 230 52.132
50 19.748 19.863 49.963 51.643 50 20.746 20.863 58.963 59.643
100 20.308 30.603 53.079 54.896 100 21.267 21.664 59.431 61.021
[10] –Muc and Chwał (2016)
observed that the critical buckling loads are higher in the uniaxial loading case.Additionally, the
buckling load coefficients increase considerably as the thickness ratio decreases. The variations
of both curves for two loading conditions are very slow above the a/h = 40 ratio (are only a
little above a/h=40).
Fig. 1. Effect of the length-to-thickness ratio on the critical buckling load for cross-ply laminates
4.3. Effect of ply orientation on the critical buckling load
The buckling load coefficient for a square and antisymmetric angle-ply laminated plate is
tested under uniaxial compressive loading; the effect of ply orientation for various numbers of
layers of the angle-ply laminate is plotted in Fig. 2. All the edges are supposed to be simply
supported, and material 5 of Table 1 is used in all cases. It is observed that in all cases, the
Fig. 2. Effect of the ply angle on the critical buckling load
Hygrothermoelastic buckling response of composite laminates... 11
critical buckling load increases at first butdecreases then.Byvarying the fiber orientation angles
from 0◦ to 90◦, it is observed that the maximum critical buckling load occurs at 45◦.
4.4. Effect of the elastic moduli ratio on the critical buckling load
The variations of critical buckling coefficients of antisymmetric cross-ply laminated plates
under uniaxial and biaxial loadings are demonstrated in Figs. 3, respectively. The results are
presented for a/h=10. It is observed that as the elastic moduli ratio rises, the critical buckling
load also increases in both uniaxial and biaxial loadings; however, in biaxial cases, the buckling
loads are approximately half of the corresponding uniaxial values at all analyzed ratios.
Fig. 3. Variation of the buckling load for square antisymmetric cross-ply laminate when a/h=10;
(a) uniaxial loading, (b) biaxial loading
4.5. Effect of thermal loads on the buckling of laminates
Buckling under thermal loads for a laminated plate consisting of 10 plies of material 4 is
examined using TSDMmodel and compared with 3D elasticity solutions. The thermal buckling
coefficients ofλT =α0Tcr areprovided inTable5.Theobtained results are in excellent agreement
with the 3D elasticity results proposed byNoor and Burton (1992), for both the thin and thick
laminated plates. In this case, the critical buckling loads correspond to the buckling modes
of m,n = 1,2, because the laminates under high temperature variations are mainly subjected
to the biaxial loading condition. The results confirm that the buckling in the thick laminated
plates occur at higher temperatures compared to the thin ones. In Fig. 4, the thermo-buckling
curve is plotted for a simply supported square and [±45◦] antisymmetric angle-ply laminate.
The obtained results by the present model are very close to the analytical solutions proposed
by Singha et al. (2001).
Table 5.Thermal buckling coefficient λT =α0Tcr for a square angle-ply laminated plate
a/h Present
3D solution
(Noor and Burton, 1992)
100.0000 0.7463 ·10−3 0.7458 ·10−3
20.0000 0.1739 ·10−3 0.1721 ·10−3
10.0000 0.5782 ·10−3 0.5820 ·10−3
6.6667 0.1029 0.1034
5.0000 0.1436 0.1515
4.0000 0.1777 0.1886
3.3333 0.2057 0.2063
12 M. Kazemi
Fig. 4. Thermo-buckling path plotted for a simply supported square [±45◦] antisymmetric
angle-ply laminate
4.6. Effect of change in moisture concentration on the buckling load
Theeffects of changes inmoisture concentrations on theuniaxial buckling load coefficientsλU
of a cross-ply [(0/90)s] laminate using material 5 is presented in Table 6. The buckling loads
are evaluated by reducing the material properties and increasing the moisture concentration.
The parameter (E2)c=0% is used to calculate the buckling load coefficient λU using the TSDM
model. In Fig. 6, the variation of the buckling load coefficient with respect to the moisture
concentration is shown for different b/h ratios. As it is seen from this figure, in thin plates, the
buckling coefficient decreases faster compared to the thick ones. However, the slope is almost
linear for both thin and thick laminates, and the thin plates may buckle due to a little change
in the moisture concentration, even in the absence of external loads.
Table 6.Effect ofmoisture concentration on the critical buckling load coefficient of a symmetric
cross-ply laminated plate for various values of the thickness-to-length ratio
a/h C [%] Present Dafedar and Desai (2002)
5
0.0 6.9932 7.1383
0.5 6.8911 7.0365
1.0 6.7963 6.9420
1.5 6.7320 6.8776
10
0.0 11.3466 11.4275
0.5 11.0183 11.0990
1.0 10.7205 10.8009
1.5 10.4631 10.5435
20
0.0 13.6835 13.7106
0.5 12.5247 12.5517
1.0 11.4879 11.5147
1.5 10.4582 10.4851
40
0.0 14.4529 14.4602
0.5 10.0180 10.0254
1.0 6.0708 6.0781
1.5 1.9523 1.9596
Hygrothermoelastic buckling response of composite laminates... 13
5. Conclusions
A new finite element formulation is developed using MHSDT for investigating the effects of
elasto-hygrothermal loads in the buckling of composite laminated plates. The transverse stresses
through thickness of a plate and the continuity of displacements are entirely satisfied in the
proposed formulation. From the extensive numerical investigation, the results obtained using
Trigonometric Shear Displacement Model (TSDM) is in excellent agreement with the three-
-dimensional elasticity solutions as well as other equivalent higher-order theories. The variations
of the critical buckling load are presented for different lamination lay-ups, elastic constants and
length-to-thickness plate ratios. The effect of thermally-induced loading and moisture concen-
tration on the buckling and post-buckling of the laminated plates are investigated using TSDM
formulation. The following conclusions are obtained:
• In the hygrothermal buckling analysis of composite plates, it is mandatory to exploit
refined higher-order theories dealing with the transverse normal deformation.
• Increasing themoisture concentrations and temperatureswould result in a reduction in the
buckling and post-buckling strength. The results also confirm that the post-buckling cha-
racteristics are significantly affected by a rise in the temperature, moisture concentration,
transverse shear deformation, plate geometry, total number of plies and fiber orientation.
• Increasing the length-to-thickness ratio, the number of layers and the orthotropic ratio
(E1/E2) would lead to an increase in the buckling strength due to in-plane compressive
loading.
• The critical buckling load is higher in the case of uniaxial loading compared to the biaxial
one.
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Manuscript received March 30, 2017; accepted for print June 7, 2017