Elsevier

Composite Structures

Volume 93, Issue 1, December 2010, Pages 153-161
Composite Structures

Nonlinear static response of laminated composite plates by discrete singular convolution method

https://doi.org/10.1016/j.compstruct.2010.06.005Get rights and content

Abstract

In this paper, large deflection analysis of laminated composite plates is analysed. Nonlinear governing equation for bending based on first-order shear deformation theory (FSDT) in the von Karman sense is presented. These equations have been solved by the method of discrete singular convolution (DSC). Regularized Shannon’s delta (RSD) kernel and Lagrange delta sequence (LDS) kernel are selected as singular convolution to illustrate the present algorithm. The effects of plate aspect ratio, fiber orientation, boundary conditions, thickness-to-side ratio, and applied load on the nonlinear static response of the laminated plate are investigated.

Introduction

Fiber reinforced composite materials are being increasingly used in the aerospace, aeronautical, mechanical and ship industry. Laminated composite structures such as beams, plates and shells are the main members of lightweight structure due to their high specific strength and low specific density. In general, linear analysis of laminated plates and shells has been used in the past. In the early studies, classical plate theory (CPT) based on Kirchhoff hypothesis are introduced for static and dynamic analyses of such structures. However, this theory is not suitable for some cases such as frequencies analysis of thick plates and postbuckling analysis of plates. By this time, different shear deformation theories for laminated plates have been developed to include the effect of transverse shear deformation [1], [2], [3], [4]. These are the classical plate theory (CPT), the first-order shear deformation theory (FSDT) and the Reddy’s higher-order shear deformation theory (HSDT). The first-order shear deformation laminated plate theory is considered in the present study. In this theory, a first-order displacement field is assumed for transverse strain through the thickness. Fundamental governing equations for nonlinear dynamic analysis of laminated plates based on von Karman’s large deflection plate theory were derived by Reddy [5] and Reddy and Chao [6]. In recent years, linear and nonlinear analysis of laminated composite plate and shell structures have found increased applications [7], [8], [9], [10], [11], [12], [13]. Nath and Shukla [14] and Shukla and Nath [15] presented some solutions for nonlinear analysis of laminated plates. Nonlinear static and dynamic response of laminated plates are also investigated by Nath et al. [16] using the Chebyshev series method. Nonlinear analysis of plates has been studied using the Chebyshev series method by Nath and Kumar [17]. A few studies concerning the nonlinear analyses of plates and shells have been carried out, namely by Civalek [18], [19], [20], [21], Kant et al. [22], [23], [24], Putcha and Reddy [25], and Zhang and Kim [26].

In the past 10 years, some new methods for numerical analysis of partial differential equations in applied mechanics have become quite popular. These are differential quadrature methods, meshless methods, and discrete singular convolution methods. Recently, the method of discrete singular convolution (DSC) proposed by Wei [27], [28] has been increasingly applied to solve many engineering and sciences problems such as mathematical physics [29], [30], [31], [32] and solid mechanics [33], [34], [35], [36], [37], [38], [39], [40], [41]. The present paper deals with the numerical solution of nonlinear static analysis problem of laminated plate by the method of DSC.

This paper is organized into six sections. In Section 1, an introduction and historical reviews are given. Governing equations are derived for nonlinear static response of laminated rectangular plates in Section 2. A general theory of DSC method is presented in Section 3. In Section 4, governing equations and solution procedures are presented in Section 4. The numerical results for static response are presented and discussed in Section 5. Finally, a conclusion is given in Section 6.

Section snippets

Governing equations

A laminated rectangular plate of dimensions a, b, and h, shown in Fig. 1 is considered. A Cartesian coordinate Oxy is located in the middle surface of the plate. The displacement field at a point in the plate using the first-order shear deformation theory is given as [15]u(x,y,z)=u0(x,y)+zψx(x,y),v(x,y,z)=v0(x,y)+zψy(x,y),w(x,y,z)=w0(x,y).where u, v and w are the displacement components of point (x, y, z), u0, v0 and w0 are the displacement components at a point on the mid-plane of the plate in x

Discrete singular convolution (DSC)

Discrete singular convolution (DSC) method is a relatively new numerical technique in applied mechanics. The method of discrete singular convolution (DSC) was proposed to solve linear and nonlinear differential equations by Wei [27].

The method of discrete singular convolution (DSC) is an effective and simple approach for the numerical verification of singular convolutions, which occur commonly in mathematical physics and engineering. The discrete singular convolution method has been extensively

Method of solution

Using DSC method to discretize the spatial derivatives in Eqs. (9a), (9b), (9c), (9d), (9e), the derivatives of the displacement components can be given by [42]D11(U)+D12(V)+D13(ΨX)+D14(ΨY)+D15(W)=0D21(U)+D22(V)+D23(ΨX)+D24(ΨY)+D25(W)=0D31(U)+D32(V)+D33(ΨX)+D34(ΨY)+D35(W)=0D41(U)+D42(V)+D43(ΨX)+D44(ΨY)+D45(W)=0D51(U)+D52(V)+D53(ΨX)+D54(ΨY)+D55(W)=0The differential operators given in the governing Eqs. (58a), (58b), (58c), (58d), (58e) and the related DSC operators are given in Appendix A.

The

Numerical results and discussions

In this subsection some numerical results have been presented. The transverse shear correction factor is considered to be 5/6. In the following solutions, three sets of composite materials, unless mentioned otherwise, are considered [14], [46], [47]:

  • Material-I

E1/E2=3,G12/E2=0.5,G23/E2=0.33,G13=G12,υ12=0.25

  • Material-II

E1/E2=25,G12/E2=0.5,G23/E2=0.2,G13=G12,υ12=0.25

  • Material-III

E1/E2=40,G12/E2=0.6,G23/E2=0.2,G13=G12,υ12=0.25Furthermore, two types of loading conditions are considered:

  • Load 1: Uniform

Conclusions

The geometrically nonlinear static analysis of laminated thick plates is numerically investigated. The analytical model is derived using the first-order shear deformation plate theory along with the von Karman nonlinear plate theory. Convergence and verification of the method are examined through comparison with linear and nonlinear results of other analytical and numerical approach. The effects of some geometric and material properties on nonlinear deflections are investigated. It is also

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