ISSN : 2288-0232(Online)

**Journal of the Korean Society for Advanced Composite Structures Vol.3 No.4 pp.17-26**

DOI : http://DOI https://doi.org/10.11004/kosacs.2012.3.4.017

# 다적층 복합면재를 갖는 비등방성 샌드위치판의 휨해석

# Bending Analysis of Anisotropic Sandwich Plates with Multi-layered Laminated Composite faces

### Abstract

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- 1. Introduction
- 2. Theoretical formulation
- 3. Governing equation
- 4. Numerical examples
- 5. Conclusions

### 1. Introduction

Recently, the composites sandwich bridge superstructures as a potential solution to the problem due to the material’s superior properties has been increased for the renewal of the deteriorated bridge with rapid construction and minimum disruption to public traffic (Ji et al., 2010). The advanced composites have advantages, such as high strength or stiffness per unit weight, fast construction and easy installation (Park et al., 2012). The sandwich plate is constructed by combining two pieces of plate with high stiffness and light one with a thick core material. The progress in material science and production techniques has led to wide application in the manufacturing of sandwich plate structures. Researches related to the sandwich structure have been conducted since 1940 (Livbove et al., 1948). In the history of theoretical analysis, the face layers of the sandwich plate were first assumed to be membranes with no bending stiffness by Reissner (Pagano, 1972), and the governing equation was derived by him. For the face plate, the classical plate theory ignored the strain of the core in the direction of the thickness in order to derive the governing equation. The effect of transverse shear strain was further considered by Whitney (1973) and Pagano (1972) by employing the Mindlin Theorem. Owing to the existence of a vast literature on isotropic faces sandwich plates, this study is limited to sandwichplates having laminated composite faces. There has been interested in the analysis of sandwich plates having generally metal, polymer laminated composite faces. A series of such analyses using the energy method was conducted by Rao and Kaeser and Rao (Mallikarjuna et al., 1993). However, they only considered shear deformation of the core.

The objective of this study is to present a formulation of governing equation for a general sandwich plates with laminated composite faces including individual effects of transverse shear deformation and bending stiffness. The obvious choice to model sandwich plates with thick laminated composite faces is a zig-zag deformation presented by Allen (1969). The present model is a development of the Allen zig-zag model (Allen, 1969). By including transverse shear deformations of the laminated composite faces, current method will be capable of analyzing thick laminated plates, general anisotropic sandwich plates. The results analyzed by present model will be compared with analytical solutions.

### 2. Theoretical formulation

The deformations through the thickness of sandwich plates with thick composite faces using the zig-zag model (Allen, 1969) are shown in Fig.1, where ( x_{ 1}, z_{ 1} ), ( x_{ 2},z _{2} ),( x_{ 3}, z_{ 3} ) are local coordinates of the upper face, core and lower face, respectively. From Fig.1, the deformations of the upper face, core and lower face can be expressed as follows:

**Fig. 1.**Cross sections of assumed deformation of the sandwich plates with composite faces (Allen, 1969)

and in the y-direction,

and in the z-direction

The same method can be used to calculate deformations in the core:

For the lower face:

where u, v, w are the displacement of the mid-surface in the x, y, z axes, and θ x , θ y are the rotation angles of the xy and yz-plane caused by flexure and t , h are the thickness of composite faces and core, while the subscripts 1, 2, 3 denote the upper face, core, and lower face, respectively. The strains in terms of the displacements are given by the usual expressions:

giving for the upper face

Similar expressions are obtained for the lower face and core. We assume that Hooke's Law is valid and the constitutive equations are given by:

in which terms are the usual stiffness coefficients.

Integrating the stresses through the thickness of each component, we get the inplane forces and moment resultants for the upper face, core, and lower face in the form:

The shear stress resultants are given by:

where [A] is the extensional stiffness matrix, [B] is the extensional-bending coupling stiffness matrix, [D] is the bending stiffness matrix.

where 1, 2, and 3 refer to upper face, core and lower face, respectively. The strain energies of the upper face, core, and lower face are given by:

where i=1,2,3 refers to upper face, core, and lower face, respectively. The corresponding shear strain energies are:

where U_{fbi} and U _{fsi} are bending and shear energies of upper face, core, and lower face. Substituting equations (2) and (3) into equations (8) and (9) we get a long expression for total strain energies of the upper face, core, and lower face. The total internal energy is then

External energies due to the lateral loads are given by the equation:

The total potential energy is the sum of internal energies and external energies:

### 3. Governing equation

Here we use the principle of virtual displacements to derive the governing equations appropriate for the displacement field in equations (1) and constitutive equation in equations (4). Therefore, the application of the principle of total potential energy gives nine governing equations (Reddy, 1984).

These nine governing equations are given below.

We will analyze a rectangular plate with length a and width b. The edges of the plate are assumed to be simply-supported such that shear deformations are prevented in the cross-sectional planes around the edges. The boundary conditions of such a plate are idealized as:

For essential boundary conditions given equation (14) above, assumed displacements are chosen in the form :

and a and b are length and width of the plate, respectively, and m and n are the half-wavelength integers. Substituting the equation (15) into the equation (13), and collecting the coefficients, one obtains:

Matrix [K] is the coefficient matrix, { δ } and {F} refer to the displacement vector and load vector.

### 4. Numerical examples

A rectangular composites sandwich plate with length a and width b will be analysed as shown Fig. 2.

**Fig. 2.**Geometry of a composites sandwich plates with laminated faces

The thickness of face and core in the Fig. 2 is t, h. The material properties of faces and core are indicated in Table 1. The problem was solved using a series which exactly satisfied the boundary conditions.

**Table 1.**Material properties of faces and core (Gpa)

For simply supported antisymmetric angle-ply faces sandwich plate as shown in Fig. 2, the laminated plate stiffness are given by below:

The analysis results presented in this study are illustrated in Table 2, and Fig. 3 for the square laminated plates with two layers [0/90]. The accuracy of the current method is ascertained by comparing the solutions from the sandwich plates theory with composite faces based on the zig - zag model (Allen, 1969) to the results of the classical laminated plates theory (CPT).

**Fig. 3.**Comparison of normalized central deflection with the side to thickness ratio (a/h) of a general laminated plates, [0/90]

**Table 2.**Comparison of normalized central deflection with side to thickness ratio (a/h) of a laminated plates,[0/90] [w_{2}E_{2}t^{3}10^{2}/(q_{2}a^{4})]

The present results considered the bending stiffness of the core and also the transverse shear deformations of the laminated faces are higher than those calculated according to the first order shear deformation theory (FSDT) (Ji et al., 1996), higher order shear deformation theory (HSDT) (Ji et al., 1998) shown in Table 2, Fig. 3. There are particularly significant difference for ratio a/h 〈 5 as shown in Table 2. Therefore, current method will be consistent with the results as apply this method for the thick laminated plates.

Results for composites sandwich plates with laminated faces are illustrated in Fig. 4.

**Fig. 4.**Comparison of central deflection with the side to thickness ratio(a/h) of a sandwich plates with composite faces, [0/CORE/0].

The analyzed results are higher than that calculated according to the first order shear deformation theory, and higher order shear deformation theory shown in Fig. 4. Table 3, and Fig. 5 show the comparison between the present theory and classical sandwich theory. The classical sandwich theory considers only transverse shear deformations of the core. The bending stiffness of the core is excluded, since it was assumed that the core is very flexible.

**Table 3.**Comparison of central deflection with the side to thickness ratio(a/h) of a sandwich plates [0/CORE/0], [w_{Z}E_{2}t^{3}10^{2}/(q_{Z}a^{4})]

**Fig. 5.**Comparison of central deflection with the side tothickness ratio(a/h) of a composites sandwich plate [0/CORE/0]

**Fig. 6.**Central deflection of composites sandwich plates with laminated faces [θ_{3}^{gr/ep}/(90-θ)^{gl/ep}/CORE/(θ-90)^{gl/ep}/θ^{gr/ep}]

Fig. 6 shows the influence of antisymmetric angle-ply laminates with ply orientations [θ/θ/θ/-θ/CORE/θ/-θ/-θ/-θ] as shown in Fig. 7.

**Fig. 7.**Typical hybrid sandwich plates with laminated facings: [θ_{3}^{gr/ep}/(90-θ)^{gl/ep}/CORE/(θ-90)^{gl/ep}/θ_{3}^{gr/ep}]

Results are presented for the composite sandwich plate of the laminated faces with four layers. The maximum value of the normalized displacement is shown to occur at the lay-up of [0/0/0/-90/core/90/0/0/0]. Fig. 8 shows the central deflection related to thickness of upper face t_{1} and the lower face t_{2}. Composites sandwich plate having symmetric angle-ply faces are superior as compared with that having anisotropic faces.

**Fig. 8.**Comparison of central deflection as to thickness variation of for the faces

The lay-up of [45/-45/core/-45/45] configuration is shown the largest value of the normalized moment M_{x} . plots as given in Fig. 9.

**Fig. 9.**Normalized moment (Mx) in y-direction with the variation of fiber angle [M_{x}/(q_{z} a^{2}]

The [90/-90/core/-90/90] configuration yields the largest value of the normalized moment M_{y} . plots are given in Fig.10.

**Fig. 10.**Normalized moment(My) in x-direction with the variation of fiber angle [M_{y}/(q_{z} b^{2}]

Fig. 11 shows the influence of antisymmetric angle-ply laminates with ply orientations ［Θ/Θ/Θ/ -Θ/ CORE /Θ/Θ/Θ/Θ］ Results are presented for the composites plate faces with four layer. The maximum value of the normalized moment M _{xy} is seen to occur at the lay-up sequence, [0/0/0/-90/core/90/0/0/0].

**Fig. 11.**Normalized moment (Mxy) in x-direction with the variation of fiber angle[M_{xy}/(q_{z} a^{2}]

### 5. Conclusions

The governing equations for bending analysis of composites sandwich plates with thick laminated faces based on zig-zag model are derived. In this study, results of the bending analysis under lateral uniform loads are shown for the composites sandwich plates with laminated faces. For the validation of this study, the present analysis was used for the general laminated plates.

The current analysis result presented in this study shows higher than one by the first order shear deformation, but smaller than one by the higher order shear deformation theory in the laminated plates. As apply the present method to investigate the bending behavior of laminated plates, it is necessary to divide the laminated plate's thickness into three components, namely upper face, core, and lower face. The thickness of each component is arbitrary. The information presented in this study might be useful to the design of the composites sandwich plates with laminated faces. And since the present analysis considers the bending stiffness of the core and the transverse shear deformations of the laminated faces, it is expected that the analysis method is capable to analyze the general laminated plates considering shear deformations.

### Reference

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