Static, free vibration and transient response of laminated composite curved shallow panel – An experimental approach

Accepted Manuscript Static, Free Vibration and Transient Response of Laminated Composite Curved Shallow Panel-An Experimental Approach Sushree S. Sahoo, Subrata K. Panda, Trupti R. Mahapatra PII:

S0997-7538(16)30043-2

DOI:

10.1016/j.euromechsol.2016.03.014

Reference:

EJMSOL 3303

To appear in:

European Journal of Mechanics / A Solids

Received Date: 9 September 2015 Revised Date:

24 March 2016

Accepted Date: 25 March 2016

Please cite this article as: Sahoo, S.S., Panda, S.K., Mahapatra, T.R., Static, Free Vibration and Transient Response of Laminated Composite Curved Shallow Panel-An Experimental Approach, European Journal of Mechanics / A Solids (2016), doi: 10.1016/j.euromechsol.2016.03.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Static, Free Vibration and Transient Response of Laminated Composite Curved Panel-An Experimental Approach

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Sushree S. Sahoo and Subrata K. Panda Department of Mechanical Engineering, National Institute of Technology, Rourkela, Odisha, PIN-769008

Experimental Model-1 Model-2 Model-3

0.8

0.6

0.2

10

20

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0.4

30

40

400 350

[±45°]s

Experimental Model-1 Model-2 Model-3 [0°/90°]s

Experimental Model-1 Model-2 Model-3

Experimental Model-1 Model-2 Model-3

300 250 200 150 100

50 0 1

50

60

2

3

4

Mode

Load

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Central Deflection (mm)

1.0

Experimental Model-1 Model-2 Model-3

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Experimental Model-1 Model-2 Model-3 [0°/90°]s

1.2

[±45°]1

[±45°]s

Natural Frequency (Hz)

[±45°]1

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In this article the free vibration, static and transient behaviour of laminated Glass/Epoxy composite panel is analyzed based on two higher-order theories and simulation model. The necessity of the higher-order model for the laminated structure is highlighted by comparing the responses with corresponding experimental analysis i.e., three point bend test and modal analysis for the static and free vibration cases.

Static comparison

Free vibration comparison

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Static, Free Vibration and Transient Response of Laminated Composite Curved Shallow Panel-An Experimental Approach

Department of Mechanical Engineering, National Institute of Technology, Rourkela, Odisha, PIN-769008 3

School of Mechanical Engineering, KIIT University, Bhubaneswar, Odisha, PIN-751024

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1,2

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Sushree S. Sahoo1, Subrata K. Panda2, Trupti R. Mahapatra3

Abstract

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In this present article, the static, free vibration and the transient behavior of the laminated composite flat/curved panels have been investigated. The deflections and the natural frequencies of the laminated composite flat panels have been computed numerically using two higher-order

mid-plane

kinematics

and

compared

with

the

available

published

numerical/analytical results. The results are also validated with the experimentally obtained

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responses (three point bend test and modal analysis). In addition to that, the transient behavior of the laminated structure is also computed numerically and compared with the results available in the open literature. The static, free vibration and the transient responses of the

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laminated panel are also computed using the simulation model developed in commercial finite

1

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element package (ANSYS) based on the ANSYS parametric design language code and

Sushree S Sahoo, Research Scholar, Department of Mechanical Engineering, National Institute of Technology Rourkela-769008, Odisha, India E-mail: [email protected] *2 Subrata K Panda, Assistant Professor, Corresponding author at: Department of Mechanical Engineering, National Institute of Technology Rourkela-769008, Odisha, India E-mail: [email protected] [email protected] *3 Trupti R Mahapatra, Assistant Professor, School of Mechanical Engineering, KIIT University, Bhubaneswar-751024, Odisha, India E-mail: [email protected]

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compared with present numerical and experimental results. Finally, the importance of the developed higher-order models and the effect of the different parameters have been shown by

Nomenclature

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computing the responses for different parameters and discussed in details.

=

Cartesian coordinate axes

(u, v, w)

=

displacements of any point along the x, y and z-direction

(u0, v0, w0)

=

displacements of a point on the mid-plane of the panel along x, y and z-

θ x ,θ y

=

rotations with respect to y and x-direction

φ x , φ y , λx , λ y ,θ z

=

higher order terms of Taylor series expansion

U

=

total strain energy

V

=

W

=

a, b and h

=

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direction

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(x, y, z)

length, breadth and thickness of a flat panel

=

radius of curvature

=

Young’s modulus in the respective material direction

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E1 , E2 and E3

work done

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R

total kinetic energy

=

Young’s modulus in a direction inclined at an angle of 450 to the x axes

G12 , G23 and G13

=

Shear modulus in their respective plane (xy, yz and xz plane)

υ12 , υ 23 and υ13

=

Poisson’s ratio in their respective direction (xy, yz and xz plane)

ρ

=

Density

E45

Keyword: Experimental analysis; Laminated curved panel; ANSYS; HSDT; Bending; Transient

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analysis; Free vibration; FEM

1. Introduction

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The use of layered composite structures in aerospace, shipbuilding, civil and mechanical structural applications has witnessed a tremendous increase in recent decades, primarily due to their innumerable attractive mechanical properties features such as light weight, high specific

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strength, high specific stiffness and formability into complex shapes. These advantageous properties have stimulated the interest of several researchers in the accurate prediction of the

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structural behavior of laminated panels. For the purpose of numerical analysis, finite element method (FEM) has been proven as a versatile tool for the analysis of complex laminated structures/structural components.

Since the last three decades, with the aim of overcoming the shortcomings of the earlier

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researchers, the study on the static, free vibration and transient analysis of laminated composite panels have been continuing to turn up with new and modified mathematical models and subsequent solution steps. Numerous researchers have already accomplished the numerical

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analysis of the laminated structures using classical theory, shear deformation theory, and refined theories as well. For more accurate and the realistic predictions, every so often these studies are

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enhanced. Some of the related studies have been discussed here to outline the knowledge gap on the available published literature in the present domain. The transient responses of laminated composite plates using a simple C0 isoparametric formulation based on higher order shear deformation theory (HSDT) kinematics were analyzed by Mallikarjuna and Kant (1988). Della and Shu (2005) studied free vibration of composite beams with two overlapping delaminations whose continuity and equilibrium conditions are

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satisfied between the adjoining regions of the beams. The spatial and temporal distributions of shear and normal interfacial stresses in a plated beam subjected to pulse loading were presented by Fallah et al. (2008). The static, free vibration and buckling behavior of the laminated

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composite and the sandwich plate were investigated by Cetković and Vuksanović (2009) using a generalized layerwise displacement model. Setoodah et al. (2009) analyzed thick composite laminates with the help of a finite element (FE) based computer code developed by coupling a

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generalized layerwise finite element model (FEM) with 3D elasticity theory. Topal and Uzman (2009) studied the optimization of the frequency of laminated skew plates based on FSDT.

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Generalized differential quadrature method was used by Hong (2010) to compute the transient response of thermal stresses and center displacement in laminated magnetostrictive plates under thermal vibration. Oktem (2012) used HSDT to present an analytical solution to the static analysis of functionally graded plates and doubly-curved shells with the fully simple supported

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boundary condition prescribed at all edges. The free vibration analysis of moderately thick symmetrically laminated general trapezoidal plates was examined by Zamani et al. (2012) with various combinations of boundary conditions obtained using first order shear deformation theory

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(FSDT). Ahmed et al. (2013) studied the static and the dynamic behavior of Graphite /Epoxy laminated composite plate under transverse load using an eight-noded isoparametric quadratic

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element in the framework of the FSDT. Transient responses and natural frequencies of sandwich beam with inhomogeneous functionally graded core were investigated by Biu et al. (2013) by proposing a novel truly meshfree method. Heshmati and Yas (2013) investigated the dynamic response of functionally graded multi-walled carbon nanotube (MWCNT) polystyrene nanocomposite beams subjected to multi-moving loads applying Newmark’s explicit integration technique. A mesh-free model was developed for dynamic analysis of functionally graded and

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uniformly distributed nanocomposite cylinders reinforced by single-walled carbon nanotubes by Moradi-Dastjerdi et al. (2013) whose effective material properties are estimated using a micromechanical model. Singh and Nanda (2013) analyzed numerically and experimentally the

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dynamic behavior of two layered and tack welded beams undergoing the mechanism of interfacial slip. Duc and Quan (2014) inspected the dynamic response of FGM double curved shallow shells resting on elastic foundations and in the thermal environment whose formulations

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are based on the classical shell theory. Kumar et al. (2014) solved the forced vibration problems of laminated composite and sandwich shells applying the first 2D FE implementation of the

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higher order zigzag theory (HOZT) incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. Versino (2014) proposed a generalized Refined Zigzag Theory to doubly-curved multilayered thin and thick structures to analyze their static behavior. Alibeigloo and Alizadeh (2015) presented static and free vibration

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behaviors of two type of sandwich plates whose governing equation of motion and constitutive relation are based on three-dimensional theory of elasticity. The FE analysis of the displacement, velocity and acceleration distributions of the unidirectional Carbon/Epoxy composite plate using

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commercial FE package ANSYS was investigated by Milan et al. (2015). Tornabene et al. (2015) examined the free vibration behavior of laminated composite thick and moderately thick elliptic

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cones, cylinders and plates employing generalized differential quadrature (GDQ) method. The dynamic behavior of singly and doubly-curved panels reinforced employing the Local Generalized Differential Quadrature (LGDQ) method to solve by Tornabene et al. (2015) numerically free vibration problems. Pandey and Pradyumna (2015) analyzed dynamic response of two types of functionally graded material sandwich plates with nonlinear temperature variation along the thickness and the FGM having temperature dependent material properties

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applying layerwise finite element formulation. Thakur and Ray (2015) developed a C0 FE model based on the HSDT for moderately thick and deep doubly curved laminated composite shells of various shapes. A simple and efficient method for precise assessment of transverse shear stresses

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in laminated composite shallow shells by using a displacement based C0 2D FE model was derived by Khandelwal and Chakrabarti (2015) from refined higher order shear deformation theory (RHSDT) and a least square error (LSE) method. Qu and Meng (2015) presented a semi-

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analytical approach for prediction of the vibration and acoustic responses of an arbitrarily shaped, multilayered shell using a higher order shear deformable zig-zag shell theory to describe

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the displacement field of the multilayered shell. Liu et al. (2016) used the differential quadrature FE method for vibration of isotropic beams, plates and layered composite plates using a layerwise theory with independent rotations in each layer. Accurate description of the multiple delaminations and transverse cracks in double-curved laminated composite shells was proposed

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by Li (2016) using the extended finite element method (XFEM) and layerwise theory (LWT) and then developed an extended layerwise method (XLWM). It is evident from the above literature review that, many numerical/analytical/experimental

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studies have already been reported to address the issues related to the static, free vibration and transient responses of laminated composite structures using the FSDT/ HSDT kinematics in

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conjunction with FEM. We note that no study has been reported yet on the static, free vibration and the transient responses of the laminated flat/curved panels numerically using the HSDT kinematics and successive experimentation. The primary objective of this article is to show the necessity of the higher-order models for the analysis of the static, free vibration and transient behavior of the laminated structure by comparing the responses with the available numerical, experimental and simulation results. In order to do so, a general mathematical model of the

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laminated composite flat/curved panel has been developed using the HSDT mid-plane kinematics and discretised using a nine noded isoparametric Lagrangian element with nine and ten degrees of freedom per node. The desired responses are obtained using the customized

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homemade computer code in MATLAB environment based on the present FE model. In addition to the above, a simulation model is also developed in commercial FE package (ANSYS) based on ANSYS parametric design language (APDL) code and compared with those available

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published literature, present numerical responses, and subsequent experimental results. Finally, the study has been extended to show the applicability of the proposed higher-order model for the

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evaluation of the static, free vibration and the transient behavior of different shell geometries (spherical, cylindrical, ellipsoid, hyperboloid and flat). The effect of different geometrical parameters (thickness ratio, aspect ratio, curvature ratio, support conditions and modular ratio) on the static response, frequency responses and the transient behavior are also computed and

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discussed in detailed.

2. Theoretical Formulations

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In the present analysis, a laminated composite panel composed of a finite number of orthotropic layers of uniform thickness is considered as in Fig. 1. The displacement field within

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the laminate is assumed to be based on the HSDT as in (Reddy, 2004), where the in-plane displacements are expanded as cubic functions of thickness coordinate while the transverse displacement varies either linearly and/or constant through the panel thickness. In addition to this the use of shear correction factor has been eliminated by considering the third-order parameters to be the function of deflection and the rotations.

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2.1. Displacement models

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In this study, three different models have been proposed and developed and utilized for further analysis. The different kinematic models developed in the present investigation are discussed in the following lines.

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Model-1

The laminated composite panel model is developed mathematically based on the higher-order

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kinematics using the displacement field as in (Reddy, 2004):

u ( x, y , z , t ) = u0 ( x, y , t ) + zθ x ( x, y , t ) + z 2φx ( x, y , t ) + z 3λx ( x, y, t ) v ( x, y , z , t ) = v0 ( x, y, t ) + zθ y ( x, y , t ) + z 2φ y ( x, y , t ) + z 3λ y ( x, y , t ) w ( x, y , z , t ) = w0 ( x, y , t )

(1.1)

where, t is the time and u, v and w are the displacements of any point along the x, y and z

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coordinate axes respectively. u0 , v0 and w0 are corresponding displacements of a point on the mid-plane and θx and θ y are the rotations of normal to the mid-surface (z = 0) about the y and x-

EP

axes, respectively. The other functions say, φ x , φ y , λ x , λ y are the higher-order terms in the

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Taylor series expansion to maintain parabolic variation. Model-2

In continuation to the earlier case, one more numerical model is also developed based on another HSDT kinematics in which the linear variation of displacement function along thickness direction is assumed as in (Singh and Panda, 2014):

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u ( x, y, z , t ) = u0 ( x, y ) + zθ x ( x, y ) + z 2φx ( x, y ) + z 3λx ( x, y ) v ( x, y, z , t ) = v0 ( x, y ) + zθ y ( x, y ) + z 2φ y ( x, y ) + z 3λ y ( x, y )

(1.2)

w ( x, y , z , t ) = w0 ( x, y ) + zθ z ( x, y )

variation of displacement function along the thickness direction. Model-3

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where, θz is the higher order terms in the Taylor series expansion which account for the linear

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As discussed earlier, a simulation model has been developed in ANSYS using APDL code.

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For the discretization purpose, Shell 281 element has been chosen from ANSYS element library. The shell element has more nodes than other shell elements and hence it is suitable for analysing thin to moderately thick shell structure. The element has eight nodes with the following displacement equations as in the FSDT:

u ( x, y, z, t ) = u0 ( x, y ) + zθ x ( x, y )

(1.3)

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v ( x, y, z, t ) = v0 ( x, y ) + zθ y ( x, y )

w ( x, y, z, t ) = w0 ( x, y ) + zθ z ( x, y )

The stress-strain relations for the kth lamina oriented at an arbitrary angle φ about any

(2)

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{σ } = Qij  {ε j }

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arbitrary axes are given by:

where, {σ } , Q ij  and {ε } are the stress matrix, reduced stiffness matrix and strain matrix, respectively. The strain matrix for any curved panel can be further written as:

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(3)

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 ∂u w     ∂x + R   x      ∂v w   +   ε    xx  ∂y R y     ε  yy      ∂u ∂v 2 w   {ε } = γ xy  =  + +   γ    ∂y ∂x R xy    xz    ∂u ∂w u   γ yz    + −    ∂z ∂x R x       ∂ v + ∂w − v    ∂z ∂y R y  

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where, Rx and R y are the radius of curvature in x and y-axis, respectively and R xy (~∞) is the twist radius.

The stress vector can be rewritten in force vector form as:

{F} = [ D] {ε }

(4)

Bij

Cij

Cij Eij

Eij Fij

Fij Wij

Eij  Fij  Wij   H ij 

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 Aij B [ D] = Cij ij   Eij

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The elements of the property matrix [ D ] are defined as:

2, ...6.

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where,  Aij , Bij , Cij , Eij , Fij , H ij , Wij  =

(5)

zk

∑ ∫ ( Q ) ( z ) dz n

m

ij

k =1 zk −1

k

for ( i, j = 1, 2,...6 ) and m = 0, 1,

2.2. Finite element formulations The displacement fields for different assumed kinematic models are expressed in terms of desired field variables and the models are discretised using suitable FEM steps.

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The displacement vector, d at any point on the mid-surface for any model is given by n

d = ∑ N i ( x, y )d i

(6)

i =1

{d i } = {u0

{d i } = {u0

i

i

v0i w0i θ xi θ yi φxi φyi λ xi λ xi

v0i w0i θ xi θ yi θ zi

}

T

} , {d } = {u T

i

0i

v0i w0i θ xi θ yi θ zi φxi φyi λ xi λ yi

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where,

}

T

and

are the nodal displacement vectors for Model-1, Model-2, and

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Model-3, respectively and Ni is the corresponding interpolating function (shape function), associated with the ith node. In this present analysis, a nine noded isoparametric quadrilateral

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Lagrangian element is utilised for the discretisation purpose and the details of the shape functions can be seen in (Cook et al., 2000). The total number of degrees of freedom per each element is eighty-one and ninety, respectively for the Model-1 and Model-2. However, the Model-3 has been discretised using the default element as in ANSYS library.

to the following form:

{ε } = [T ]{ε }

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The strain matrix is written in the matrix form after introducing the FEM steps and conceded

(7)

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where, {ε } and [T] are the mid-plane strain and the thickness coordinate matrix and the mid-

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plane strain vector can be further reduced as:

{ε } = [ BL ]{d}

(8)

where, [ BL ] is a general strain displacement relation matrix. The total strain energy of the laminate can be expressed as:

U=

+ h /2  1  T  ∫ {ε } {σ } dz dxdy ∫∫ 2  − h /2 

(9)

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Eq. (9) can be rewritten by substituting strains and stresses as:

(

)

1 T {ε } [ D]{ε } dxdy ∫∫ 2

where, [ D ] =

+ h /2

∫ [T ]

T

(10)

Qij  [T ] dz

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U=

− h /2

The kinetic energy of the laminate can be expressed as:

{ } {δ&} dV

1 ρ δ& ∫ 2V

T

(11)

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Te =

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where, ρ and {δ&} are the mass density and the global velocity vector, respectively. The total work done by an externally applied load, F is given by W = ∫ {δ }

T

{ F } dA

A

2.3. Governing equations

(12)

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2.3.1. Free vibration analysis

The final form of governing equation of free vibrated composite flat panel is obtained by

t2

δ ∫ (T − U )dt = 0

(13)

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t1

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using Hamilton’s principle and expressed as:

Substituting Eqs. (6), (10) and (11) into Eq. (13), the final form of the equation will be conceded as:

[ M ]{d&&i } + [ K ]{di } = 0

(14)

where, d&&i is the acceleration, di is the displacement and [K] and [M] are the stiffness and mass matrices, respectively which can be further expressed as:

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[ K ] = ∫ [ BL ] [ D ][ BL ] dA T

A

(15)

[ M ] = ∫ [ N ] [ N ] ρ dA T

A

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Neglecting the required matrices, the eigenvalue form of the governing equation to obtain the natural frequency of the system is conceded as:

([ K ] − ω [ M ]) ∆ = 0 2

(16)

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where, ω and ∆ are the natural frequency and the corresponding eigenvector, respectively.

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2.3.2. Static analysis

The final form of governing equation for static analysis of flat panel is obtained using variational principle:

δ ∏= δU −δW = 0

(17)

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where, δ is the variational symbol and ∏ is the total potential energy. The equilibrium equation for the static analysis is obtained by substituting Eq. (17) for Eqs. (6), (10) and (12) as follows:

(18)

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[ K ] {d } = { F }

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The Eqs. (15) and (18) are solved by using the following sets of support conditions to avoid any rigid body motion as well as reduce the number of unknowns. Simply supported (SSSS):

v = w = θ = θ = φ = λ = 0 at x=0 and a; 0 0 y z y y u = w = θ = θ = φ = λ = 0 at y=0 and b 0 0 x z x x

(19)

Clamped (CCCC):

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u = v = w = θ = θ = θ = φ = φ = λ = λ = 0 at x=0 and a; y=0 and b; 0 0 0 x y z x y x y

(20)

Free (FFFF): 0

= v = w = θ = θ = θ = φ = φ = λ = λ = 0 at x=0 and a; y=0 and b; 0 0 x y z x y x y

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u

Hinged (HHHH): u 0 = v 0 = w0 = θ z = φ y = λ y = 0

at x=0 and a;

(22)

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2.3.3. Transient Analysis

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u0 = v0 = w0 = θ z = φx = λx = 0 at y=0 and b;

(21)

In the transient analysis, at a particular time t, static equilibrium along with the effects of acceleration dependent inertia forces and velocity-dependent damping forces are considered. The equation of equilibrium governing the linear transient response of a system is expressed as

[ M ]d&& + [C ]d& + [ K ]d = [ F ]

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follows:

(23)

where, [ M ] , [C] and [ K ] are the mass, damping and stiffness matrices, respectively. [ F ] is the

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external force vector and d&& , d& and d are the acceleration, velocity and displacement vectors,

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respectively. Now, the transient equation is solved for few time step ∆ t within the whole time period of T, using the constant average acceleration steps of Newmark’s integration scheme. The desired transient equation is obtained using Newmark’s integration parameters such as α , δ and

a0 to a7 as in (Bathe, 1982). For a particular time step t, the effective stiffness matrix is expressed as follows:

[ Kˆ ] = [ K ] + a0 [ M ]

(24)

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Similarly, the effective load matrix for the successive time step t + ∆ t is taken to be: t + ∆t

[ Fˆ ] =

t + ∆t

[ F ] + [ M ](a0 t d + a2 t d& + a3 t d&&) + [C ](a1 t d − a4 t d& + a5 t d&&)

(25)

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Now the desired displacement, acceleration and velocity from the transient analysis can be computed using the following equations: [ Fˆ ]

t + ∆t

d&& = a0 ( t + ∆t d − t d ) − a 2 t d& − a3 t d&&

t + ∆t

d& = t d& + a6 t d&& + a7 t + ∆t d&&

3. Results and Discussion

(26)

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t + ∆t

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[ Kˆ ] t + ∆ t d =

The static, free vibration and the transient responses of the laminated composite flat/curved panels are computed using the developed FE code in MATLAB environment. In order to overcome the effects of shear locking and membrane locking in the present analysis selective

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integration method has been adopted with respect to the bending and shear terms. The validity of the present higher-order models is examined by comparing the responses with those available published literature, simulation (ANSYS) and subsequent experimental results. The effect of

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various parameters (shell configurations, thickness ratios, aspect ratios, modular ratios, curvature

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ratios and support conditions) on the static, free vibration and the transient responses are computed using the present HSDT models and discussed in detailed. Here, in this analysis different sets of composite material properties are utilized for the computational purpose and presented in Table 1. The composite material properties are also obtained experimentally and utilised further in the numerical analysis. In addition, few properties are taken from the earlier published literature for the comparison and the parametric study as well. The material properties of three different laminations ([±450], [±450]s and [00/900]s) of

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Glass/Epoxy composite flat panels are evaluated experimentally and presented in Table 1 namely, M4, M5 and M6, respectively. The longitudinal Young’s modulus is obtained through the unidirectional tensile test using the universal testing machine (UTM, INSTRON 1195) at

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National Institute of Technology (NIT), Rourkela, Odisha, India. Similarly, three more pieces are also prepared from the same specimen by taking the longitudinal, transverse and the inclined (angle of inclination 45° to the longitudinal direction) piece, to compute the shear modulus. The

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specimen dimensions are prepared in accordance to ASTM standard (D 3039/D 3039M). The properties of the Glass/Epoxy laminated composite plates are obtained in the said UTM by

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setting the loading rate as 1 mm/minute. The UTM and the deformed specimen configurations are shown in Fig. 2(a) and (b), respectively. It is important to mention that the Poisson’s ratio for the present computational purpose is taken to be 0.17 as in (Crawley, 1979). Similarly, the shear modulus of the individual specimen has been computed based on the data obtained from the

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 experimentation and the given formula as in (Jones, 1975):  G = 12   

1

4 1 1 2υ − − − 12 E 45 E1 E 2 E1

   and   

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the constants are defined in the nomenclature. 3.1. Convergence and comparison study

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The convergence behavior of the static responses of the laminated composite panels has been established by considering the flat panel example available in Xiao et al. 2008. The desired responses have been computed using the present higher-order numerical model (Model-1 and Model-2) and the ANSYS simulation model (Model-3) and presented in Fig. 3(a), (b) and (c), respectively. The nondimensional static deflections are computed for three ([0˚/90˚/0˚]) and four ([0˚/90˚]s) layer cross-ply laminated composite flat panels under uniformly distributed load

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(UDL). The nondimensional central deflections are obtained for three different thickness ratios (h/a= 0.05, 0.1 and 0.2). The geometrical parameter, material properties and support condition are taken same as to the considered reference (Xiao et al. 2008). The nondimensionalisation of

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the central deflection is computed using the formula: w = (100 E2 h 3 q0 a 4 ) w , where, w is the maximum central deflection. It is clearly observed that the present results are converging well with mesh refinement. Based on the convergence behavior, a (6×6) mesh has been used for

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further computation. The results are computed using the proposed models and compared with

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various published results (Reddy 2004, Xiao et al. 2008 and Belinha and Dinis 2006) and presented in Table 2. It is clear from the table that the present numerical results are in good agreement with the previous analytical and numerical results.

Now the developed models (Model-1, Model-2 and Model-3) are extended to show the convergence behavior of free vibration responses. In order to do so, simply supported two

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layered (00/900) and four layered ([00/900] s) square laminated composite cross-ply flat panels with four modular ratios (E1/E2=3, 10, 15 and 30) are considered for computation. The geometrical parameters, material properties (M2 material properties as in Table 1) and support

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conditions are taken to be the same as in Matsunaga, 2004. The responses computed using

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Model-1, Model-2 and Model-3 are presented in Fig. 4(a), (b) and (c), respectively. For the present study, the nondimensional frequency is obtained using the formulae: ω = ω h ρ E 2 , where ω is the natural frequency. It is evident from the figure that the present results are converging well with mesh refinement and a (6×6) mesh is sufficient to compute the responses further. The present responses are also compared with those available published results (Noor and Burton 1989, Kant and Kommineni 1994 and Matsunaga 2004) and presented in Table 3. It

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can also be observed that the present results are showing good agreement with that to the analytical and numerical results. Similarly, the transient responses are validated by comparing the results for two examples

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(two layer and eight layer) of cross-ply square simply supported laminated composite plate using the geometrical parameters and material properties (a =0.25m, h =0.01m and M3 material properties as in Table 1) same as to the reference (Maleki et al. 2012). For the comparison

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purpose the results are computed using all three models (Model-1, Model-2 and Model-3) and presented in Fig. 5(a) and (b). The transient responses are computed under uniform step load, q0

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by setting the time steps as 1 µs. The nondimensional deflections are computed using the formula: w = (100 E2 h3 q0 a 4 ) w where, w is the central deflection. The present responses are compared with that of the reference values (Reddy 2004 and Maleki et al. 2014). It can be clearly

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observed that the results are showing very good agreement with the published results. 3.2. Experimental static and free vibration study

In order to build more confidence on the presently developed numerical model, the responses

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are further compared with the experimental results. The bending responses of three different specimens of three lamination schemes ([±450], [±450]s and [00/900]s) of Glass/Epoxy composite

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plate are obtained experimentally using UTM (INSTRON 5967 with 30 kN load cell) available at NIT Rourkela, Odisha, India, with M4, M5 and M6 material properties as presented in Table 1. For the three-point bend test, two opposite edges of the plate were held in simply supported condition. The three point bending test set up and the deformed shapes of all the three specimens are presented in Fig. 6(a) and (b), respectively. The specimens for the experimental analysis have been prepared as per ASTM standard (D 3039/D 3039M). The recommended loading rate for the

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analysis is fixed as 2 mm/minute. The experimental results are further compared with all the numerical and simulation results obtained using Model 1, Model 2 and Model 3 and presented in Table 4. It is interesting to note that both the HSDT models are showing very good agreement

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with the corresponding experimental results. This indicates the necessity of the present HSDT type model for the accurate analysis of laminated structures under large deformation regime. In addition, the experimental flexural stress-strain diagram of Glass/Epoxy laminated composite

[±450] s and [00/900]s) during the three-point bend test.

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specimens are also presented in Fig. 7(a)-(c) for all three types of lamination scheme ([±450],

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Further, the modal test has been conducted to obtain the free vibration responses using PXIe1071 (1) (National Instruments) available at NIT Rourkela, Odisha, India and shown in Fig. 8(a). The PXIe-1071 is an eight-slot chassis that features a high-bandwidth backplane to meet a wide range of high-performance test and measurement application needs which is ideal for processor-

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intensive, deterministic modular instrumentation and data acquisition real-time applications. The vibration responses are recorded for all the three types of Glass/Epoxy composite plates (3) under CFFF support (5) through the accelerometer (4) mounted on the plate. The plate is excited

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with the help of an impact hammer (6) on any arbitrary points and the signal is captured through the accelerometer. The accelerometer is a type of sensor that sense the vibration and convert it

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into the analogue voltage signal and it again convert to digital form through an inbuilt analogue to digital converter within the PXIe. These signals produced by the sensors due to the impact are received at the controller through the PXIe and further processed with the help of LABVIEW software. In LABVIEW, a virtual instrument (VI) program circuit has been developed for the recording of experimental data. The VI is mainly comprised of three components: a back panel, a front panel, and a connector panel. The front panel is built using the controls and the indicators.

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The controls are the inputs and allow a user to supply the information to the VI and the indicators are the output which display (2) the results based on the inputs given in the VI. The back panel is a block diagram comprises the graphical source code and it contains structure and function. It

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performs the operation on the controls and the supply data to the indicator. Finally, the value of the responses were displayed on the computer screen. In this experiment, the acceleration of Glass/Epoxy flat panels is recorded with the assistance of PXIe by exciting the plate with the

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help of an impact hammer. Fig. 8(b) depicts the block diagram of the LABVIEW software for the recording of the signal and subsequent analysis. Finally, the velocity and the displacement

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will be evaluated from the recorded acceleration as shown in the LABVIEW circuit using the single and double integration blocks. Moreover, the acceleration signals are passed through the power spectrum module of the LABVIEW specifically for fast Fourier transformation to obtain the frequency domain responses. The peaks of the frequency response spectrum give the natural

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frequencies of vibration for different modes

Finally, the natural frequencies obtained from the experimentation are compared with those numerical results computed using the proposed higher-order model and the simulation model as

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well. The comparison study presented in Table 5 clearly indicates the necessity of present HSDT model i.e., the Model-1 and Model-2 instead of the FSDT model as adopted in the Model-3 for

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the analysis of laminated structures. In addition to this, the mode shapes of the first five natural frequencies of symmetric angle-ply [±450]s Glass/Epoxy laminated flat panel are computed and presented in Fig. 9(a)-(e). It is true that the mode shapes show the direction of vibration only but not give any idea related to the numerical value. It can be clearly seen that the present responses are within the expected line. 3.3. Numerical illustrations

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Based on the present convergence and the comparison study, it is understood that both the proposed HSDT models are capable of solving the static, free vibration and the transient responses of the laminated composite structures accurately. It is also observed that the responses

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computed using the Model-1 are closer to the experimental and numerical results in most of the cases. Hence, some new illustrations for different geometrical and material parameters are computed using the Model-1 to highlight the quantitative understanding on the static, free

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vibration and the transient behavior of the laminated composite flat/curved panel structures. The responses are computed for different parameters such as the thickness ratios, the aspect ratios,

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the modular ratios, the curvature ratios, the support conditions and the shell geometries and discussed in details.

3.3.1. Effect of aspect ratio on static response

It is true that the stiffness and the stability of any structural component are highly dependent

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on the aspect ratio (a/b), mainly in the case of laminated structures. The nondimensional central deflection of four layered simply supported laminated cross-ply composite flat panels subjected to UDL with varying thickness ratios (a/h=5, 10, 15, 20 and 25) and M1 material properties are

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analyzed by employing the Model-1. The static responses of the flat panel are computed for different aspect ratio (a/b=1, 1.5, 2, 2.5 and 3) and M1 composite properties and are presented in

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Fig. 10. It is clear from the figure that the nondimensional central deflection parameters tend to decrease as the aspect ratio increases and the results are within the expected line. 3.3.2. Effect of modular ratio on static response

In this example, the influence of modular ratio on the nondimensional central deflection of a square four layered simply supported cross-ply laminated composite flat panels subjected to UDL with various thickness ratios (a/h=5, 10, 15, 20 and 25) and M1 material properties is

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analyzed. The responses are computed for five modular ratios (E1/E2 = 10, 15, 20, 25 and 30) and presented in Fig. 11. The responses indicate that the nondimensional central deflection of the flat panel decreases as the modular ratio decreases. It is due to the fact that with the increase in

of the plate directly. 3.3.3. Deformation shape of various shell geometry

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modular ratio, the longitudinal Young’s modulus increases which in turn increases the stiffness

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It is well known that the manmade shells are classified based on their geometry rather than their load bearing capacity and also their deflection parameters largely depend on the

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geometrical configurations. In this example, the effect of different geometrical configurations (spherical, ellipsoid, cylindrical, hyperboloid, and flat) on the nondimensional central deflection of square four layered simply supported cross-ply laminated composite panels under UDL with a/h= 20, R/a = 5 and M1 material properties is examined and plotted in Fig. 12 (a)-(e). It can be

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clearly observed that the maximum and minimum deflections are obtained for the flat and the spherical panels, respectively.

3.3.4. Effect of support condition on in-plane normal stresses and interlaminar shear stress

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From the design point of view, the stress is one of the important parameters in the laminated structure and it has also great physical significance in accordance to the support

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condition. Therefore, in this example the in-plane normal stresses (σxx and σyy) and the inter-laminar shear stresses (τxz, τyz and τxy) are computed using the Model-1 for square four layered cross-ply laminated composite flat panel under UDL with a/h= 20 and M1 material properties. The responses are computed for five different support conditions (CFCF, CCCC, SSSS, SCSC and HHHH) and presented in Fig. 13(a) - (e) with respect to nondimensionalised thickness coordinate (z/h). It can be clearly observed that both the

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normal and shear stresses are higher for two support conditions (SSSS and HHHH) and lower for the CCCC support. It is well known that as the deflection and stresses of any

are following the same trend. 3.3.5. Effect of thickness ratio on free vibration response

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structure increases or decreases in accordance to their number of constraint and the results

The thickness ratio of the laminated panel is inversely proportional to the stiffness of the

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structure. It is also true that the structure becomes thin as the thickness ratio increases and the stiffness of the structure decrease subsequently. The frequency of the structure is the function of

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the stiffness and the mass matrices. In this example, the effect of thickness ratio on the frequency parameter has been analyzed by computing the responses through the Model-1. The nondimensional fundamental frequency responses of simply supported two layered cross-ply laminated composite flat panel for five aspect ratios (a/b= 1, 1.5, 2, 2.5 and 3) and five thickness

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ratios (a/h = 20, 25, 30, 35 and 40) are computed using the M2 material properties and E1/E2=30. The nondimensional frequencies are presented in Fig. 14. It is clearly observed that the nondimensional frequency decreases as the thickness ratio increases irrespective of the aspect

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ratio and the responses are within the expected line. 3.3.6. Effect of curvature ratio and shell geometries on free vibration response

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The different forms of the shell structure, i.e. from deep to shallow, is defined by its curvature ratio (R/a). It is also true that the stretching energy is high in comparison to the bending energy for deep shell geometries and it subsequently affect the final responses of the structure as well. The nondimensional fundamental frequency parameters of two layered square simply supported composite shell panel of different geometries (spherical, ellipsoid, cylindrical and hyperboloid) are computed using the Model-1 and other parameters as E1/E2=30, a/h=5, and M2 material

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properties. The frequencies are computed for five curvature ratios (R/a = 10, 20, 30, 40 and 50) and presented in Fig. 15. It is interesting to note that the nondimensional frequency parameters decrease as the curvature ratio increases for each case of the shell geometries. The responses are

3.3.7. Effect of support condition on free vibration response

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decreases progressively in descending order of hyperbolic, cylindrical, elliptical and spherical.

The type of support condition or constraint plays the significant role on the stiffness

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characteristics of any structure/structural component. In any structure, the stiffness increases as the number of constraint increases or the degree of freedom decreases and this may affect the

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natural frequency of the panel noticeably. In this example, the effect of various type of support conditions on the nondimensional natural frequency of two-layer composite flat panels is examined based on the Model-1. The responses are computed for five support conditions (CCCC, SCSC, HHHH, SSSS and CFFF), five aspect ratios (a/b=1, 1.5, 2, 2.5 and 3) with other

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parameter as a/h=5, E1/E2=30 and M2 material properties and presented in Fig. 16 has been examined. It is interesting to note that the responses are following a decreasing trend from CCCC, SCSC, HHHH, SSSS and CFFF.

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3.3.8. Effect of modular ratio on transient response

The transient behavior of two-layer cross-ply square simply supported composite flat panel is

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investigated in this example. The nondimensional central deflections with respect to time are computed using the Model-1 and presented in Fig. 17. The responses are obtained for five different modular ratios (E1/E2=30, 35, 40, 45 and 50) with a/h=25 and M3 material properties under uniform step load ( q0 ). It is observed that the nondimensional central deflections with respect to the time are decreased as the modular ratio increases and the responses are within the expected line.

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3.3.9. Effect of shell geometry on transient response

In this example, the transient behavior of two layered cross-ply square simply supported composite laminated panel is investigated for various shell geometries (spherical, cylindrical,

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ellipsoid, hyperboloid and flat). The nondimensional central deflections of the laminated panel are computed using the Model-1 under uniform step load ( q0 ) by setting R/a=5, a/h =25, and M3 material properties and presented in Fig. 18. It is clearly observed that the maximum and

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minimum responses are obtained for the flat and the spherical panel, respectively. It is also observed that the hyperboloid and flat panel responses are close to each other for few case.

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3.3.10. Effect of aspect ratio on transient response

The nondimensional central deflections of two layer cross-ply simply supported composite laminated flat panel is computed in this example for five aspect ratio (a/b= 1, 1.25, 1.5, 1.75 and 2) by employing Model-1. The responses are computed for the flat panel case by taking a/h=25

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and M3 material properties and presented in Fig. 19. It is clear from the figure that, the nondimensional deflection parameters are decreasing as the aspect ratio increases due to the

(

)

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4. Conclusion

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nondimensional factor as in the nondimensional form  w = 100 E2 h3 q0 a 4 w .

The static, free vibration and the transient responses of the laminated composite flat/curved Glass/Epoxy composite panels have been computed numerically using two proposed HSDT kinematic model through a homemade computer code developed in MATLAB. Initially, the responses are compared with the published numerical/analytical results and further compared with experimental responses obtained from the bending (three-point bend test) and the vibration (modal analysis) cases. In addition to the above, the validity of the present higher-order models

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is also presented by comparing the responses with the simulation model developed in ANSYS using APDL code. The composite properties are obtained experimentally through the uniaxial tensile test (UTM, INSTRON 1195) and utilized further for the experimental and numerical

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analysis of the responses. The convergence and the validation behavior of the present higherorder models have been checked by solving various numerical examples for thick and thin laminated shell panel. Based on the comparison study, it is understood that the Model-1 show

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closer values with the experiments for both the static and free vibration cases. Finally, some new

following conclusions are obtained.

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examples have been solved using the Model-1 for different geometrical parameters and the

a. It is clear from the comparison study that, the HSDT model is inevitable for the analysis of the laminated flat/curved panel structure.

b. The present results also indicate that the different panel configurations have the

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considerable effect on their responses.

c. It is also observed that the minimum and maximum nondimensional central deflections are obtained for the spherical and the flat panels, respectively.

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d. It is understood that the thickness ratios, the aspect ratios, the curvature ratios and the support conditions have also a considerable effect on the static, free vibration and

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transient responses of the flat/curved laminated panel structure.

e. Finally, it is observed that the Model-1 and Model-2 both are showing the good capability of solving the vibration, bending and transient responses. But it is interesting to note that the vibration responses computed using the Model-1 are close to the experimental results. However, the Model-2 is capable of solving the bending

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responses of the laminated structure more accurately due to the absence of the unrealistic assumption of stretching term through the thickness.

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Acknowledgments

This work is under the project sanctioned by the Department of Science and Technology (DST) through grant SERB/F/1765/2013-2014 Dated: 21/06/2013. Authors are thankful to

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DST, Govt. of India for its consistent support.

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Table 1 Material Properties.

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Table 2 Comparison study of nondimensional deflections of simply supported laminated

composite flat panel under UDL.

supported cross-ply laminated flat panel.

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Table 3 Comparison study of nondimensional fundamental frequency of square simply

Table 4 Experimental comparison of central deflections of Glass/Epoxy laminated composite

flat panel under point load.

Figure captions

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Table 5 Natural frequency of cantilever Glass/Epoxy laminated composite flat panels.

Fig. 1. Geometry of a laminated doubly curved panel. Fig. 2(a). UTM INSTRON 1195.

Fig. 2(b). Glass/Epoxy laminated composite specimen after tensile test.

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Fig. 3(a)-(c). Convergence study of nondimensional deflection of a simply supported three and

four layered cross-ply laminated flat panel.

Fig. 4(a)-(c). Convergence study of nondimensional frequency of a simply supported two and

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four layered cross-ply laminated flat panel.

panel.

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Fig. 5(a). Nondimensional deflection of a simply supported two layered cross-ply laminated flat

Fig. 5(b). Nondimensional deflection of a simply supported eight layered cross-ply laminated

flat panel.

Fig. 6(a). Three point bend test set up (INSTRON 5967). Fig. 6(b). Deformed Glass/Epoxy laminated composite specimen after three point bend test.

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Fig. 7(a)-(c). Flexural stress-strain diagram of three laminations ([±450]1, [±450]s) of

Glass/Epoxy composite specimens.

composite flat panel 4.Accelorometer 5.Fixture 6.Impact hammer.

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Fig. 8(a). Experimental set up 1.NI PXIe-1071 2.Computer screen 3.Glass/Epoxy laminated

Fig. 8(b). Block diagram of the LABVIEW for experimental data recording.

Fig. 9(a)-(e). Mode shapes of first five natural frequencies of cantilever symmetric angle-ply

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Glass/Epoxy laminated composite flat panels.

cross-ply laminated composite flat panel.

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Fig. 10. Effect of aspect ratio on the nondimensional deflection of four layered simply supported

Fig. 11. Effect of modular ratio on the nondimensional deflection of square four layered simply

supported cross-ply laminated composite flat panel.

Fig. 12(a)-(e). Deformation shape of various shell geometry of square four layered simply

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supported cross-ply laminated composite panels.

Fig. 13(a)-(e). Effect of varying support condition on in-plane and shear stresses of a square four

layered cross-ply laminated composite flat panel.

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Fig. 14. Effect of varying thickness ratio on nondimensional frequency of two layered simply

supported composite laminated flat panel.

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Fig. 15. Effect of varying curvature ratio on nondimensional frequency of two layered simply

supported composite laminated panels. Fig. 16. Effect of varying boundary condition on nondimensional frequency of two layered

composite laminated panels. Fig. 17. Effect of modular ratio on the nondimensional deflection of a simply supported two

layered cross-ply laminated flat panel.

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Fig. 18. Effect of shell geometry on the nondimensional deflection of a simply supported two

layered cross-ply laminated panel. Fig. 19. Effect of aspect ratio on the nondimensional deflection of a simply supported two

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layered cross-ply laminated flat panel.

Material Properties. MATERIAL -1 [M1]

MATERIAL -2 [M2]

E1

250GPa

Open

E2

10GPa

1GPa

E3

10GPa

1GPa

G12

5GPa

0.5GPa

G13

5GPa

G23

2GPa

υ12

0.25

υ13

0.25

υ 23

0.25

ρ

-

MATERIAL -4 [M4]

MATERIAL -5 [M5]

MATERIAL -6 [M6]

525GPa

4.669GPa

4.408GPa

5.639

21GPa

4.351GPa

4.081GPa

4.926

525GPa

4.351GPa

4.081GPa

4.926

10.5GPa

3.25GPa

1.1GPa

0.75

10.5GPa

3.25GPa

1.1GPa

0.75

0.35GPa

4.2GPa

1.625GPa

0.55GPa

0.375

0.3

0.25

0.17

0.17

0.17

0.3

0.25

0.17

0.17

0.17

0.3

0.25

0.17

0.17

0.17

1kg/m3

800kg/m3

1900kg/m3

1900kg/m3

1900kg/m3

EP

0.5GPa

AC C

Table 2

MATERIAL -3 [M3]

TE D

Material Properties

M AN U

Table 1

Comparison study of nondimensional deflections of simply supported laminated composite flat panel under UDL. Nondimensional Deflection

34

ACCEPTED MANUSCRIPT

0

0 /90 /0

0

Model2

Model3

2.069 1.0863 0.7866 2.0582 1.1027 0.8024

1.9344 1.032 0.771 1.995 1.026 0.7912

2.464 1.168 0.7964 2.328 1.149 0.804

2.1644 1.086 0.7688 2.1496 1.109 0.7844

HSDT TPS Xiao et al. (2008) 2.1556 1.08 0.7613 2.14 1.0955 0.7775

Reddy (2004)

Belinha and Dinis (2006)

1.0219 0.7572 1.025 0.7694

1.0225 0.7583 1.0248 0.7698

SC

(00/900)s

0.2 0.1 0.05 0.2 0.1 0.05

Model -1

EP

TE D

M AN U

Table 3 Comparison study of nondimensional fundamental frequency of square simply supported cross-ply laminated flat panel. Nondimensional Frequency Lamination Modular Noor and Kant and Model- Model- ModelMatsunaga Scheme Ratio Burta Kommineni 1 2 3 (2002) (1989) (1994) 3 0.2327 0.2445 0.2387 0.2392 0.2388 0.2389 10 0.2629 0.2705 0.2654 0.2671 0.2675 0.2669 (00/900)1 15 0.2791 0.2853 0.2785 0.2815 0.2809 0.2812 30 0.3156 0.3184 0.3057 0.3117 0.3117 0.3116 3 0.2445 0.2554 0.2472 0.2493 0.2495 0.2491 10 0.3089 0.312 0.3019 0.3063 0.3002 0.3063 (00/900)s 15 0.3374 0.3375 0.3255 0.3307 0.3306 0.3309 30 0.3882 0.3824 0.3657 0.3726 0.3725 0.3731

AC C

0

Thickness Ratio

HSDT MQ Xiao et al. (2008)

RI PT

Lamination Scheme

3D FEM Xiao et al. (2008) 2.3218 1.1541 0.7951 2.3218 1.1541 0.7951

Table 4 Experimental comparison of central deflections of Glass/Epoxy laminated composite flat panel under point load.

Lamination Scheme

0

[±45 ]1

Load (in N)

Experimental

15 25 35

0.24951 0.44312 0.65591

Central deflection (mm) Model-1 Model-2 0.2725 0.4542 0.6356

35

0.2800 0.4667 0.6531

Model-3 0.2800 0.4380 0.6140

ACCEPTED MANUSCRIPT

0.8399 1.0263 0.2847 0.4739 0.6634 0.8531 0.9487 0.1667 0.2779 0.3890 0.5001 0.6112

0.7890 0.9640 0.2620 0.4340 0.6050 0.7790 0.8770 0.1550 0.2580 0.3610 0.4640 0.5670

RI PT

0.8174 0.9988 0.2702 0.4499 0.6297 0.8098 0.8998 0.1600 0.2667 0.3733 0.4799 0.5865

SC

[00/900]s

0.89071 1.16734 0.2759 0.4498 0.6283 0.8019 0.8895 0.1795 0.2893 0.4038 0.5160 0.6275

M AN U

[±450]s

45 55 15 25 35 45 55 15 25 35 45 55

Table 5

Experimental comparison study of natural frequencies of cantilever Glass/Epoxy laminated

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

AC C

[±450]1

Mode No.

EP

Lamination Scheme

TE D

composite flat panels.

[±450]s

[00/900]s

Experimental Results 16 42 114 129.5 158 38.5 111 248 322 384 45 86.5 290 345 350.5

36

Natural Frequency (in Hz) Model-1

Model-2

Model-3

17.8525 42.1443 115.085 136.3641 157.293 39.00986 109.4509 240.5947 324.0557 383.6592 46.85928 84.65459 291.9063 348.661 352.3057

18.1316 42.3353 117.0108 138.0511 158.9004 39.26233 108.9257 239.8626 327.8117 381.8289 46.62386 83.81106 289.7259 349.791 351.7646

17.757 41.702 111.01 134.29 152.59 38.799 108.02 234.89 318.89 374.04 49.42 85.769 301.89 333.54 360.52

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

Fig. 2. Geometry of a laminated doubly curved panel.

Fig. 2(a). UTM INSTRON 1195.

Fig. 2(b). Glass/Epoxy laminated composite specimen after tensile test.

37

ACCEPTED MANUSCRIPT

MODEL-1 0 0 0 0 0 /90 /0 0 /90 /90 /0 h/a=0.2 h/a=0.2 h/a=0.1 h/a=0.1 h/a=0.05 h/a=0.05

2.0

1.5

1.0

0.5 4x4

5x5

6x6

7x7

8x8

MESHSIZE

(a)

1.5

1.0

0.5

9x9

3x3

4x4

5x5

6x6

7x7

8x8

9x9

MESHSIZE

(b)

MODEL-3 0 0 0 0 0 0 0 0 /90 /0 0 /90 /90 /0 h/a=0.2 h/a=0.2 h/a=0.1 h/a=0.1 h/a=0.05 h/a=0.05

3.0

2.5

TE D

NONDIMENSIONAL DEFLECTION

2.0

M AN U

3x3

2.5

RI PT

2.5

MODEL-2 0 0 0 0 0 0 0 0 /90 /0 0 /90 /90 /0 h/a=0.2 h/a=0.2 h/a=0.1 h/a=0.1 h/a=0.05 h/a=0.05

3.0

0 0

SC

0

NONDIMENSIONAL DEFLECTION

NONDIMENSIONAL DEFLECTION

3.0

2.0

AC C

EP

1.5

1.0

0.5 2x2

4x4

6x6

8x8

10x10

MESH SIZE

(c)

Fig. 3(a)-(c). Convergence study of nondimensional deflection of a simply supported three and four layered cross-ply laminated flat panel.

38

0.45 0.40 0.35 0.30 0.25 0.20 2x2

3x3

4x4

5x5

6x6

MESHSIZE (a) NONDIMENSIONAL FREQUENCY

0.50

0.50

MODEL-2 0 0 0 0 0/90 0/90/0 /90 Et/El=3 Et/El=3 Et/El=10 Et/El=10 Et/El=15 Et/El=15 Et/El=30 Et/El=30 0

0.45 0.40 0.35 0.30 0.25

7x7

0.20

0

RI PT

MODEL-1 0 0 0 0 0 0 /90 0 /90 /0 /90 Et/El=3 Et/El=3 Et/El=10 Et/El=10 Et/El=15 Et/El=15 Et/El=30 Et/El=30 0

SC

NONDIMENSIONAL FREQUENCY

0.50

M AN U

NONDIMENSIONAL FREQUENCY

ACCEPTED MANUSCRIPT

2x2

3x3

4x4

5x5

6x6

7x7

MESHSIZE (b)

MODEL-3 0 0 0 0 0 0 0 /90 0 /90 /0 /90 Et/El=3 Et/El=3 Et/El=10 Et/El=10 Et/El=15 Et/El=15 Et/El=30 Et/El=30

TE D

0.45 0.40 0.35

EP

0.30 0.25

AC C

0.20

2x2

4x4

6x6

8x8

10x10

12x12

MESHSIZE

(c)

Fig. 4(a)-(c). Convergence study of nondimensional frequency of a simply supported two and four layered cross-ply laminated flat panel.

39

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 5(a). Nondimensional deflection of a simply supported two layered cross-ply laminated

AC C

EP

TE D

flat panel.

Fig. 5(b). Nondimensional deflection of a simply supported eight layered cross-ply laminated

40

ACCEPTED MANUSCRIPT

Fig. 6(a). Three point bend test set up (INSTRON 5967).

M AN U

SC

RI PT

flat panel.

Fig. 6(b). Deformed Glass/Epoxy laminated composite specimen after three point bend

AC C

EP

TE D

test.

41

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

(a)

(b)

42

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

(c)

Fig. 7(a)-(c). Flexural stress-strain diagram of Glass/epoxy laminated

AC C

EP

TE D

composite specimens.

43

ACCEPTED MANUSCRIPT

Fig. 8(a). Experimental set up 1.NI PXIe-1071 2.Computer screen 3.Glass/Epoxy laminated

M AN U

SC

RI PT

composite flat panel 4.Accelorometer 5.Fixture 6.Impact hammer.

x 10

TE D

Fig. 8(b). Block diagram of the LABVIEW for experimental data recording.

-5

9

7 6

4 3 2 1 0 0

0

-1

-2 0

0

0.5 0.2

0.4

0.6

0.8

3

1

AC C

5

-5

2

EP

8

x 10

1

1

-3 0

a

0.5 0.2

0.4

0.6

b

b

44

0.8

1

1

a

ACCEPTED MANUSCRIPT

Natural frequency=39.00986

Natural frequency=109.4509 (b)

x 10

M AN U

SC

RI PT

(a)

x 10

-6

2

8 6

1.5

4

1

2

TE D

0 -2 -4 -6

0

-8

0.2

0.4

b

0.6

EP

0.5

0.8

1

a

0.5

0

-0.5 0 -1 0

1

0.5 0.2

0.4

0.6

0.8

1

b

AC C

-10 0

-5

Natural frequency=240.5947

Natural frequency=324.0557

(c)

(d)

45

1

a

ACCEPTED MANUSCRIPT

x 10

-6

8 6

RI PT

4 2 0

-4

SC

-2

0

0.5

-6 0

0.2

0.4

0.8

1

1

M AN U

0.6

a

b

Natural frequency=383.6592 (e)

Fig. 9(a)-(e). Mode shapes of first five natural frequencies of cantilever symmetric angle-ply Glass/Epoxy

AC C

EP

TE D

laminated composite flat panels.

46

2.0

MODEL-1

a/b=1 a/b=2 1.5

a/b=1.5 a/b=2.5

RI PT

a/b=3

1.0

0.5

0.0 5

10

15

20

25

M AN U

THICKNESS RATIO

SC

NONDIMENSIONAL DEFLECTION

ACCEPTED MANUSCRIPT

Fig. 10. Effect of aspect ratio on the nondimensional deflection of four layered simply

TE D

supported cross-ply laminated composite flat panel.

3.0

2.6

MODEL-1

E1/E2=10 E1/E2=20

2.4

E1/E2=15 E1/E2=25 E1/E2=30

2.2 2.0

AC C

EP

NONDIMENSIONAL DEFLECTION

2.8

1.8 1.6 1.4 1.2 1.0 0.8 0.6

5

10

15

20

25

THICKNESS RATIO

Fig. 11. Effect of modular ratio on the nondimensional deflection of square four layered simply supported cross-ply laminated composite flat panel.

47

ACCEPTED MANUSCRIPT

-3

x 10

0.8 0.6 0.4 0.2

0.05

0.1

0.15

0.2

0.15

0.2

a=20cm

0.1

0.05

0

b=20cm

X: 0.1 Y: 0.1 Z: 0.001189

1 0.8 0.6 0.4 0.2 0 0

0.05

0.1

0.15

0.2

0.2

(a)

TE D

M AN U

a=20cm

EP

0 0

ELLIPTICAL

1.2

RI PT

X: 0.1 Y: 0.1 Z: 0.00112

1

-3

SC

NONDIMENSINAL DEFLECTION

SPHERICAL

1.2

AC C

NONDIMENSIONAL DEFLECTION

x 10

48

(b)

0.15

0.1

b=20cm

0.05

0

ACCEPTED MANUSCRIPT

-3

x 10

CYLINDRICAL

1.2

X: 0.1 Y: 0.1 Z: 0.001244

1 0.8 0.6 0.4 0.2

0.05

0.1

0.15

0.2

0.1

0.15

0.2

a=20cm

0.05

1.4

HYPERBOLOID

1.2

X: 0.1 Y: 0.1 Z: 0.001281

1 0.8 0.6 0.4 0.2 0 0

0.05

0

0.4

TE D

1.2

0.6

0.2

0.15

0.1

0.05

0

b=20cm

(d)

FLAT

X: 0.1 Y: 0.1 Z: 0.001284

0.2

EP

NONDIMENSIONAL DEFLECTION

-3

1.4

0.8

0.2

M AN U

(c)

1

0.15

a=20cm

b=20cm

x 10

0.1

SC

0 0

-3

RI PT

1.4

NON-DIMENSIONAL DEFLECTION

NONDIMENSIONAL DEFLECTION

x 10

0 0

0.05

0.1

0.15

0.2

a=20cm

0.2

0.15

0.1

0.05

0

AC C

b=20cm

(e) Fig. 12(a)-(e). Deformation shape of various shell geometry of square four layered simply supported cross-

49

ACCEPTED MANUSCRIPT

RI PT

MODEL-1 CFCF CCCC SCSC SSSS HHHH

6

4

SC

2

0

-2

-4

-6 -0.6

-0.4

M AN U

INPLANE NORMAL STRESS (σxx)

ply laminated composite panels.

-0.2

0.0

0.2

0.4

THICKNESS COORDINATE (z/h)

AC C

EP

TE D

(a)

50

0.6

ACCEPTED MANUSCRIPT

MODEL-1 CFCF SCSC CCCC SSSS HHHH

100

RI PT

50

0

-50

-100

-150 -0.4

-0.2

0.0

0.2

0.4

0.6

M AN U

-0.6

SC

NORMAL INPLANE STRESS (σyy)

150

THICKNESS COORDINATE (z/h) (b)

2000

EP

SHEAR STRESS (τxz)

4000

MODEL-1 CFCF HHHH CCCC SSSS SCSC

TE D

6000

0

AC C

-2000 -4000 -6000

-0.6

-0.4

-0.2

0.0

0.2

THICKNESS COORDINATE (z/h)

(c)

51

0.4

0.6

ACCEPTED MANUSCRIPT

6000

MODEL-1 CCCC SCSC CFCF SSSS HHHH

2000

RI PT

SHEAR STRESS (τyz)

4000

0

-2000

-6000 -0.6

-0.4

-0.2

0.0

SC

-4000

0.2

0.4

0.6

M AN U

THICKNESS COORDINATE (z/h)

(d)

5

MODEL-1 HHHH SCSC CFCF CCCC SSSS

3 2

TE D

1 0 -1 -2 -3

EP

SHEAR STRESS (τxy)

4

-4 -5

AC C

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

THICKNESS COORDINATE (z/h)

(e)

Fig. 13(a)-(e). Effect of varying support condition on in-plane and shear stresses of a square four layered cross-ply laminated composite flat panel.

52

ACCEPTED MANUSCRIPT

MODEL-1

a/h=20 a/h=25 a/h=30 a/h=35 a/h=40

0.10

RI PT

0.08

0.06

0.04

0.02

0.00 1.0

1.5

2.0

2.5

3.0

M AN U

ASPECT RATIO

SC

NONDIMENSIONAL FREQUENCY

0.12

Fig. 14. Effect of varying thickness ratio on nondimensional frequency of

0.3170 0.3168 0.3166

MODEL-1 Hyperboloid Cylindrical Ellipsoid Spherical

0.3164

EP

NONDIMENSINAL FREQUENCY

0.3172

TE D

two layered simply supported composite laminated flat panel.

0.3162

AC C

0.3160 0.3158 0.3156

10

20

30

40

50

CURVATURE RATIO

Fig. 15. Effect of varying curvature ratio on nondimensional frequency of two layered simply supported composite laminated panels.

53

1.0

RI PT

MODEL-1 CFFF SSSS HHHH SCSC CCCC

1.2

0.8

0.6

SC

0.4

0.2

0.0 1.0

M AN U

NONDIMENSIONAL FREQUENCY

ACCEPTED MANUSCRIPT

1.5

2.0

2.5

3.0

ASPECT RATIO

Fig. 16. Effect of varying boundary condition on nondimensional frequency

TE D

of two layered composite laminated panels.

3.5

EP

3

E1/E2=20 E1/E2=25 E1/E2=30 E1/E2=35 E1/E2=40

2.5

2

AC C

NONDIMENSIONAL DEFLECTION

4

1.5

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5

TIME (in secs)

54

0.6

0.7

0.8

0.9

1 x 10

-3

ACCEPTED MANUSCRIPT

Fig. 17. Effect of modular ratio on the nondimensional deflection of a simply supported two

RI PT

layered cross-ply laminated flat panel.

Flat Hyperboloid Ellipsoid Cylindrical Spherical

3

SC

2.5

2

1.5

1

0.5

0 0

0.1

0.2

0.3

M AN U

NONDIMENSIONAL DEFLECTION

3.5

0.4

0.5

TE D

TIME (in secs)

0.6

0.7

0.8

0.9

1 x 10

-3

Fig. 18. Effect of shell geometry on the nondimensional deflection of a simply supported two

AC C

EP

layered cross-ply laminated panel.

55

ACCEPTED MANUSCRIPT

a/b=1 a/b=1.25 a/b=1.5 a/b=1.75 a/b=2

3

RI PT

2.5

2

1.5

1

0.5

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

M AN U

0

SC

NONDIMENSIONAL DEFLECTION

3.5

TIME (in secs)

0.8

0.9

1 x 10

-3

Fig. 19. Effect of aspect ratio on the nondimensional deflection of a simply supported two

AC C

EP

TE D

layered cross-ply laminated flat panel.

56

ACCEPTED MANUSCRIPT

Highlight 1. In this article the static, free vibration and transient behavior of laminated composite flat/curved panel have been analyzed using two higher-order shear deformation theories and simulation model developed in ANSYS.

RI PT

2. The developed models are validated by comparing the responses with those available published literature and subsequent experiments (three point bend test and modal analysis) using the experimentally evaluated laminated properties. 3. The necessity of the higher-order model for the analysis of laminated composite

AC C

EP

TE D

M AN U

SC

is highlighted based on the experimental validation.