parametric instability of square laminated plates in hygrothermal

Hindawi Publishing CorporationJournal of StructuresVolume 2013, Article ID 492839, 6 pageshttp://dx.doi.org/10.1155/2013/492839

Research ArticleParametric Instability of Square Laminated Plates inHygrothermal Environment

Manoj Kumar Rath and Shishir Kumar Sahu

Department of Civil Engineering, National Institute of Technology, Rourkela, Orissa 769008, India

Correspondence should be addressed to Shishir Kumar Sahu; [email protected]

Received 29 April 2013; Accepted 8 July 2013

Academic Editor: Mustafa Kemal Apalak

Copyright © 2013 M. K. Rath and S. K. Sahu. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The present paper investigates the parametric instability of square laminated plates subjected to periodic dynamic loadings inhygrothermal environment. The effects of various parameters like the increase in static load factor and the degree of orthotropyof simply supported composite plates at elevated temperatures and moisture concentrations on the principal instability regionsare investigated using finite element method. The effects of transverse shear deformation and rotary inertia are used to studythe antisymmetric angle-ply square plates. A simple laminated plate model is developed for the parametric instability of squarelaminated plates subjected to hygrothermal loading. A computer program based on FEM inMATLAB environment is developed toperform all necessary computations.The results show that instability of square laminated plates occurs for different parameters withan increase in temperature and moisture environment. The onset of instability occurs earlier, and the width of dynamic instabilityregions increases with a rise in temperature and moisture for different parameters.The effect of damping shows that there is a finitecritical value of dynamic load factor for each instability region below which the square laminated plates cannot become unstable.

1. Introduction

There is a tremendous increase in the utilization of com-posite materials in thin-walled structural components ofhigh speed aircrafts, submarines, automobiles and otherhigh-performance application areas. When exposed to hightemperature and moisture, the changes in vibration andstatic and dynamic stability characteristics have necessitateda strong need to understand their dynamic behavior underdifferent loading conditions.

L. W. Chen and Y. M. Chen [1] studied the free vibrationof the laminated rectangular composite plate exposed tosteady state hygrothermal environment using finite elementmethod. Sai Ram and Sinha [2] investigated the effects ofmoisture and temperature on the free vibration of laminatedcomposite plates using finite element method. Huang et al.[3] discussed the effects of hygrothermal conditions on thedynamic response of shear deformable laminated plates rest-ing on elastic foundations using amicro-to-micromechanicalanalytical model. Thangaratnam et al. [4] studied the buck-ling analysis of composite laminates for critical temperature.

The mathematical formulation is based on linear theory andthe finite element method using semiloof elements. Ram andSinha [5] investigated the effects of moisture and temperatureon the static stability of laminated composite plates. Themathematicalmodel is based on finite elementmethodwhichtakes transverse shear deformation into account. Patel et al.[6] studied the hygrothermal buckling effects on the struc-tural behavior of thick composite laminates using higher-order theory. The analysis is carried out employing a C0QUAD-8 isoparametric higher-order finite element.

Few studies are available on the behavior of compositeplates under ambient temperature and moisture subjectedto in-plane loads. Srinivasan and Chellapandi [7] studiedthe dynamic stability of rectangular plates due to periodicin-plane load by using finite strip method. Lien-Wen andJenq-Yiing [8] investigated the dynamic stability of laminatedcomposite plates by Galerkin finite element method. Kwon[9] examined the dynamic instability of layered compositeplates by finite element method by using higher-order bend-ing theory. Balamurugan et al. [10] studied the nonlineardynamic instability of laminated composite plates using finite

2 Journal of Structures

element model. Patel et al. [11] investigated the dynamicinstability of laminated composite plates supported on elasticfoundations and subjected to periodic in-plane loads, usingC1 eight-nodded shear-flexible plate element. Sahu andDatta[12] presented the dynamic stability behavior of laminatedcomposite curved panels subjected to in-plane static andperiodic compressive loads using finite element method. Theboundaries of instability region were obtained using Bolotin’smethod and were represented in amplitude-excitation fre-quency plane.

2. Mathematical Formulation

The mathematical formulation for parametric instabilitybehavior of laminated composite plates subjected tomoistureand temperature is presented. Consider a laminated plate ofuniform thickness “𝑡” consisting of a number of thin laminae,each of whichmay be arbitrarily oriented at an angle “𝜃” withreference to the 𝑥-axis of the coordinate system as shown inFigures 1 and 2.

2.1. Governing Equations. The governing differential equa-tions for vibration of a shear deformable laminated compositeplates and shells in general are specified here, but the scopeof the analysis is for composite plates. The woven fiberlaminated composite plates in hygrothermal environmentderived on the basis of first order shear deformation theoryand (FSDT) subjected to in-plane loads are employed, and ashear correction factor of 5/6 is included for all the numericalcomputations as follows:

𝜕𝑁𝑥

𝜕𝑥+

𝜕𝑁𝑥𝑦

𝜕𝑦−1

2(1

𝑅𝑦

−1

𝑅𝑥

)

𝜕𝑀𝑥𝑦

𝜕𝑦+𝑄𝑥

𝑅𝑥

+

𝑄𝑦

𝑅𝑥𝑦

= 𝑃1

𝜕2𝑢

𝜕𝑡2+ 𝑃2

𝜕2𝜃𝑥

𝜕𝑡2

𝜕𝑁𝑥𝑦

𝜕𝑥+

𝜕𝑁𝑦

𝜕𝑦+1

2(1

𝑅𝑦

−1

𝑅𝑥

)

𝜕𝑀𝑥𝑦

𝜕𝑥+

𝑄𝑦

𝑅𝑦

+𝑄𝑥

𝑅𝑥𝑦

= 𝑃1

𝜕2V

𝜕𝑡2+ 𝑃2

𝜕2𝜃𝑦

𝜕𝑡2,

𝜕𝑄𝑥

𝜕𝑥+

𝜕𝑄𝑦

𝜕𝑦−𝑁𝑥

𝑅𝑥

−

𝑁𝑦

𝑅𝑦

− 2

𝑁𝑥𝑦

𝑅𝑥𝑦

+ 𝑁𝑎

𝑥

𝜕2𝑤

𝜕𝑥2

+ 𝑁𝑎

𝑦

𝜕2𝑤

𝜕𝑦2+ 𝑁𝑎

𝑥𝑦

𝜕2𝑤

𝜕𝑥𝜕𝑦= 𝑃1

𝜕2𝑤

𝜕𝑡2,

𝜕𝑀𝑥

𝜕𝑥+

𝜕𝑀𝑥𝑦

𝜕𝑦− 𝑄𝑥= 𝑃3

𝜕2𝜃𝑥

𝜕𝑡2+ 𝑃2

𝜕2𝑢

𝜕𝑡2,

𝜕𝑀𝑥𝑦

𝜕𝑥+

𝜕𝑀𝑦

𝜕𝑦− 𝑄𝑦= 𝑃3

𝜕2𝜃𝑦

𝜕𝑡2+ 𝑃2

𝜕2V

𝜕𝑡2,

(1)

where 𝑁𝑥, 𝑁𝑦, and 𝑁

𝑥𝑦are the in-plane stress resultants,

𝑀𝑥,𝑀𝑦, and𝑀

𝑥𝑦are moment resultants and 𝑄

𝑥and 𝑄

𝑦are

transverse, shear stress resultants. 𝑅𝑥, 𝑅𝑦, and 𝑅

𝑥𝑦identify

y

x

21

b

a

𝜃

Figure 1: Arbitrarily oriented laminated plate.

z k−1

z k

21

n

kz

Figure 2: Geometry of an 𝑛-layered laminate.

the radii of curvatures in the 𝑥 and 𝑦 directions and radiusof twist.

(𝑃1, 𝑃2, 𝑃3) =

𝑛

∑

𝑘=1

∫

𝑧𝑘

𝑧𝑘−1

(𝜌)𝑘(1, 𝑧, 𝑧

2) 𝑑𝑧, (2)

where 𝑃1, 𝑃2, and 𝑃

3are the applied in-plane load.

The self-explanatory figure in Figure 3.

2.2. Dynamic Stability Studies. The equation of motion forvibration of a laminated composite panel in hygrothermalenvironment, subjected to generalized in-plane load. 𝑁(𝑡)can be expressed in the matrix form as follows:

[𝑀] { ̈𝑞} + [[𝐾𝑒] − 𝑁 (𝑡) [𝐾𝑔]] {𝑞} = 0. (3)

“𝑞” is the vector of degrees of freedoms (𝑢, V, 𝑤, 𝜃𝑥, 𝜃𝑦). The

in-plane load “𝑁(𝑡)” may be harmonic and can be expressedin the following form:

𝑁(𝑡) = 𝑁𝑠+ 𝑁𝑡cosΩ𝑡, (4)

where𝑁𝑠is the static portion of load𝑁(𝑡),𝑁

𝑡is the amplitude

of the dynamic portion of 𝑁(𝑡), and Ω is the frequency ofthe excitation. The stress distribution in the panel may beperiodic. Considering the static and dynamic components ofload as a function of the critical load, as gets

𝑁𝑠= 𝛼𝑁cr, 𝑁

𝑡= 𝛽𝑁cr, (5)

where 𝛼 and 𝛽 are the static and dynamic load factors,respectively. Using (5), the equation of motion for panel inhygrothermal environment under periodic loads in matrixform may be obtained as follows:

[𝑀] { ̈𝑞} + [[𝐾𝑒] − 𝛼𝑁cr [𝐾𝑔] − 𝛽𝑁cr [𝐾𝑔] cosΩ𝑡] {𝑞} = 0.(6)

Journal of Structures 3

x

y

z

a

bRx

Ry

Figure 3: Laminated composite curved panels under in-planeharmonic loading under hygrothermal environment.

Equation (6) represents a system of differential equationswith periodic coefficients of the Mathieu-Hill type. Thedevelopment of regions of instability arises from Floquet’stheorywhich establishes the existence of periodic solutions ofperiods 𝑇 and 2𝑇. The boundaries of the primary instabilityregions with period 2𝑇, where 𝑇 = 2𝜋/Ω, are of practicalimportance, and the solution can be achieved in the form ofthe trigonometric series

𝑞 (𝑡) =

∞

∑

𝑘=1,3,5,...

[{𝑎𝑘} sin(𝑘Ω𝑡

2) + {𝑏

𝑘} cos(𝑘Ω𝑡

2)] .

(7)

Putting this in (6) and if only the first term of the series isconsidered, equating coefficients of sin(Ω𝑡/2) and cos(Ω𝑡/2),(6) reduces to

[[𝐾𝑒] − 𝛼𝑃cr [𝐾𝑔] ±

1

2𝛽𝑃cr [𝐾𝑔] −

Ω2

4[𝑀]] {𝑞} = 0.

(8)

Equation (8) represents an eigenvalue problem for knownvalues of 𝛼, 𝛽, and 𝑃cr. The two conditions under the plusand minus signs correspond to two boundaries (upper andlower) of the dynamic instability region.The above eigenvaluesolution give of Ω, which gives the boundary frequenciesof the instability regions for the given values of 𝛼 and 𝛽.In this analysis, the computed static buckling load of thepanel is considered as the reference load. Before solving theabove equations, the stiffness matrix [𝐾] is modified throughimposition of boundary conditions.

2.3. Constitutive Relation. The constitutive relations for theplate subjected to moisture and temperature are

{𝐹} = [𝐷] {𝜀} − {𝐹𝑁} , (9)

where

{𝐹} = {𝑁𝑥, 𝑁𝑦, 𝑁𝑥𝑦,𝑀𝑥,𝑀𝑦,𝑀𝑥𝑦, 𝑄𝑥, 𝑄𝑦}𝑇

,

{𝐹𝑁} = {𝑁

𝑁

𝑥, 𝑁𝑁

𝑦, 𝑁𝑥𝑦,𝑀𝑁

𝑥,𝑀𝑁

𝑦,𝑀𝑁

𝑥𝑦, 0, 0}𝑇

,

{𝜀} = {𝜀𝑥, 𝜀𝑦, 𝛾𝑥𝑦, 𝐾𝑥, 𝐾𝑦, 𝐾𝑥𝑦, 𝜑𝑥, 𝜑𝑦}𝑇

,

(10)

Table 1: Elastic moduli of glass fiber/epoxy lamina at differenttemperatures. 𝛼

1= −0.3×10

−6/∘K, 𝛼2= 28.1×10

−6/∘K, 𝛽1= 0, 𝛽

2=

0.44.

Elastic moduli Temperature in (K)300 325 350 375 400 425

𝐸1

7.9 7.6 7.1 6.7 6.5 6.3𝐸2

7.4 6.8 6.4 6.2 5.9 5.7𝐺12

2.9 2.6 2.3 2.1 1.8 1.6]12

0.4 0.43 0.41 0.35 0.36 0.35

where𝑁𝑥,𝑁𝑦, and𝑁

𝑥𝑦are in-plane internal stress resultants,

𝑀𝑥, 𝑀𝑦, and 𝑀

𝑥𝑦are internal moment resultants, 𝑄

𝑥, 𝑄𝑦

are transverse shear resultants, 𝑁𝑁𝑥, 𝑁𝑁𝑦, and 𝑁𝑁

𝑥𝑦are in-

plane non-mechanical stress resultants due to moisture andtemperature,𝑀𝑁

𝑥,𝑀𝑁𝑦, and𝑀𝑁

𝑥𝑦= non-mechanical moment

resultants due to moisture and temperature, 𝜀𝑥, 𝜀𝑦, and 𝛾

𝑥𝑦=

in-plane strains of the midplane, 𝐾𝑥, 𝐾𝑦, and 𝐾

𝑥𝑦are

curvature of the plate, and 𝜑𝑥, 𝜑𝑦are Shear rotations in 𝑥𝑧

and 𝑦𝑧 planes, respectively.Element stiffness matrix is given by

[𝐾𝑒] = ∫

+1

−1

∫

+1

−1

[𝐵]𝑇[𝐷] [𝐵] |𝐽| 𝑑𝜉𝑑𝜂. (11)

The geometric stiffnessmatrix due to residual stresses is givenby

[𝐾𝑟

𝐺𝑒] = ∫

+1

−1

∫

+1

−1

[𝐺]𝑇[𝑆] [𝐺] |𝐽| 𝑑𝜉𝑑𝜂. (12)

The geometric stiffness matrix due to applied in-plane loadsis given by

[𝐾𝑎

𝐺𝑒] = ∫

+1

−1

∫

+1

−1

[𝐻]𝑇[𝑃] [𝐻] |𝐽| 𝑑𝜉𝑑𝜂. (13)

The element load vector due to the hygrothermal forces andmoments is given by

{𝑃𝑁

𝑒} = ∫

+1

−1

∫

+1

−1

[𝐵]𝑇{𝐹𝑁} |𝐽| 𝑑𝜉𝑑𝜂. (14)

3. Results and Discussion

The geometrical and material properties of the laminatedcomposite plates are 𝑎 = 𝑏 = 0.235m, ℎ = 0.006m (unlessotherwise stated). The material properties obtained fromtensile testing of glass/epoxy composite plates at differenttemperatures and moisture are as shown in Tables 1 and 2.

The nondimensional excitation frequency Ω =

Ω𝑎2√(𝜌/𝐸

22ℎ2) is used throughout the dynamic instability

studies, whereΩ is the excitation frequency in radian/second.The principal instability regions of woven fiber laminatedcomposite plates subjected to in-plane periodic loads areplotted with non-dimensional frequency Ω/𝜔 (ratio ofexcitation frequency to the free vibration frequency) versusthe dynamic in-plane load 𝛽. Here, a static load factor 𝛼 = 0.2


Table 2: Elastic moduli of glass fiber/epoxy lamina at differentmoisture concentrations. 𝛼

1= −0.3 × 10

−6/∘K, 𝛼2= 28.1 ×

10−6/∘K, 𝛽1= 0, 𝛽

2= 0.44.

Elastic moduli Moisture concentration in %0.0 0.25 0.5 0.75 1.0

𝐸1 7.9 7.6 7.5 7.3 7.2𝐸2 7.4 7.4 7.3 7.1 7.0𝐺12 2.9 2.9 2.8 2.7 2.6

]12 0.4 0.4 0.4 0.39 0.39

0

0.2

0.4

0.6

0.8

1

9 10 11 12

Dyn

amic

load

fact

or (𝛽

)

Nondimensional excitation frequency (Ω)

𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1

Figure 4: Variation of instability regions with temperature at 325Kfor simply supported SSSS of [45/−45]

4, woven fiber laminated

composite plates.

is taken for parametric study of laminated composite platesin hygrothermal environment throughout the analysis unlessotherwise stated.

The effect of increase in static load factor of eight-layeredantisymmetric woven fiber laminated composite plates onnon-dimensional excitation frequency is analyzed in Figures3, 4, and 5 with the increase of temperatures of 325K, 375K,and 425K, respectively.

It is observed that with the increase of static load factorfrom 0.2 to 1, the onset of dynamic instability occurs earlier,and the width of dynamic instability region also increases.

The variations of dynamic instability regions withincrease in static in-plane loads of woven fiber compositeplates with increase in moisture concentrations of 0.25%,0.5%, and 1% are observed as shown in Figures 6, 7, and8 respectively. The onset of instability occurs earlier withincrease of compressive static in-plane load; the instabilityregion tends to shift to lower frequencies and wider.

The effect of the degree of orthotropy is examined forthe eight-layered anti-symmetric angle-ply laminated plateson the non-dimensional excitation frequency is presented inFigures 9, 10, and 11 with increase of temperature of 325K,375K, and 425K, respectively. It is seen that with increase inthe values of E

1/E2=10, 20, and 40, the onset of instability

occurs earlier with decrease in degree of orthotropy. Thewidth of instability zones decreases with the increase of thedegree of orthotropy in higher temperature environment.

The variations of dynamic instability regions withincrease in the values of degree of orthotropy E

1/E2= 10,

0

0.2

0.4

0.6

0.8

1

6 7 8 9

Dyn

amic

load

fact

or (𝛽

)


𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1



composite plates.

0

0.2

0.4

0.6

0.8

1

0 2 4 6

Dyn

amic

load

fact

or (𝛽

)


𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1



composite plates.

9 9.5 10 10.5 11Nondimensional excitation frequency (Ω)

𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1

0

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)

Figure 7: Variation of instability regions with moisture concentra-tion at 0.25% for simply supported SSSS of [45/−45]

4, woven fiber

laminated composite plates.

20, and 40 are presented as shown in Figures 12, 13, and 14with increase in moisture concentration of 0.25%, 0.5%, and1% respectively. The onset of instability occurs earlier withdecrease in degree of orthotropy, with increase in orthotropythe narrow instability regions show more stiffness in highermoisture concentration environment.

Journal of Structures 5

6 7 8 9Ω)Nondimensional excitation frequencies (

𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1

0

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)


4, woven fiber


0 2 4 6Nondimensional excitation frequency (Ω)

𝛼 = 0.4𝛼 = 0.6

𝛼 = 0.8𝛼 = 1

0

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)

Figure 9: Variation of instability regions with moisture concen-tration at 1% for simply supported SSSS of [45/−45]

4, woven fiber


The onset of instability for woven fiber composite issmaller than unidirectional composite because the materialsproperties are exist in two directions due to bending stretch-ing coupling effect (Figure 15).

4. Conclusions

A parametric instability study of woven fiber laminatedcomposite plates in hygrothermal environment subjected toperiodic in-plane loads is examined. The following observa-tion can be made.

(i) The excitation frequencies of square laminated platesdecrease with the increase of temperature and mois-ture concentration due to reduction of stiffness for alllaminates.

(ii) The greater is the static load factor, the higher isthe instability region of square laminated plates inhygrothermal environment.

(iii) The increase in the degree of orthotropy is to enhancethe dynamic stability strength and to shift the occur-rence of instability to lower values of excitationfrequencies in hygrothermal environment.

0 20 40 600

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)

Nondimensional excitation frequencies (Ω)

E1/E2 = 10

E1/E2 = 20

E1/E2 = 40


4, 𝛼 = 0.2, woven fiber


0 20 40 600

0.2

0.4

0.6

0.8

1D

ynam

ic lo

ad fa

ctor

(𝛽)


E1/E2 = 10

E1/E2 = 20

E1/E2 = 40




0 20 40 600

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)


E1/E2 = 10

E1/E2 = 20E1/E2 = 40





0 20 40 600

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)


E1/E2 = 10

E1/E2 = 20

E1/E2 = 40


4,𝛼 = 0.2, woven

fiber laminated composite plates.

0 20 40 600

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)


E1/E2 = 10

E1/E2 = 20

E1/E2 = 40


4, 𝛼 = 0.2, woven


0 20 40 600

0.2

0.4

0.6

0.8

1

Dyn

amic

load

fact

or (𝛽

)


E1/E2 = 10

E1/E2 = 20

E1/E2 = 40

Figure 15: Variation of instability regions with moisture concentra-tion at 1% for simply supported SSSS of [45/−45]

4, 𝛼 = 0.2, woven


(iv) The excitation frequencies of laminated fiber compos-ite panels decrease with increase of temperature andmoisture concentration due to reduction of stiffnessfor all laminates.

(v) The high-modulus-fiber material plate is most stablein hygrothermal environment.

From the present studies, it is concluded that the parametricinstability of woven fiber laminated composite plates is influ-enced by the static load factor and degree of orthotropic. Sucha property can be utilized to tailor the design of woven fiberlaminated composite plates in hygrothermal environment.

References

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