Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5698351, 12 pages

http://dx.doi.org/10.1155/2016/5698351

## A Corotational Formulation for Large Displacement Analysis of Functionally Graded Sandwich Beam and Frame Structures

Department of Solid Mechanics, Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

Received 15 March 2016; Accepted 26 May 2016

Academic Editor: Zhiqiang Hu

Copyright © 2016 Dinh Kien Nguyen and Thi Thom Tran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A corotational finite element formulation for large displacement analysis of planar functionally graded sandwich (FGSW) beam and frame structures is presented. The beams and frames are assumed to be formed from a metallic soft core and two symmetric functionally graded skin layers. The Euler-Bernoulli beam theory and von Kármán nonlinear strain-displacement relationship are adopted for the local strain. Exact solution of nonlinear equilibrium equations for a beam segment is employed to interpolate the displacement field for avoiding the membrane locking. An incremental-iterative procedure is used in combination with the arc-length control method to compute the equilibrium paths. Numerical examples show that the proposed formulation is capable of evaluating accurately the large displacement response with just several elements. A parametric study is carried out to highlight the effect of the material distribution, the core thickness to height ratio on the large displacement behaviour of the FGSW beam, and frame structures.

#### 1. Introduction

Large displacement analysis of structures is an important topic in the field of structural mechanics. This topic grows in importance due to the development of new materials which enable structures to undergo large deformation. Many investigations on the large displacement analysis of structures using both analytical and numerical methods are reported in the literature. Numerical methods, especially the finite element method with its versatility in spatial discretization, are an effective tool for the large displacement analysis of structures.

In order to analyse beam and frame structures undergoing large displacements by the finite element method, a nonlinear beam element which can model accurately the nonlinear behaviour of a structure is necessary to formulate. Various nonlinear beam elements for analysis of planar beam and frame structures are available in the literature. Depending on the choice of reference configuration, the nonlinear beam elements can be classified into three types: the total Lagrangian formulation, the updated Lagrangian formulation, and the corotational formulation. In the corotational formulation, which will be discussed herein, the kinematics are described in an element attached local coordinate system. The finite element formulation is firstly formulated in the local system and then transformed into a global system with the aid of transformation matrices. The elements proposed by Hsiao et al. [1, 2], Meek and Xue [3], Pacoste and Eriksson [4], and Nguyen [5] are some amongst the other corotational beam elements for large displacement analysis of planar beams and frames.

Analyses of structures made of functionally graded materials (FGMs) have been extensively carried out since these materials were created by Japanese scientists in 1984 [6]. The finite element analysis is often employed to handle the complexities arising from the material inhomogeneity. Chakraborty et al. [7] proposed a finite element formulation for studying the thermoelastic behaviour of shear deformable FGM beams. The formulation using the exact solution of an equilibrium FGM Timoshenko beam to interpolate the displacement field is free of shear locking. Based on the higher-order shear deformation beam theory, Kadoli et al. [8] investigated the static behaviour of FGM beams under ambient temperature. The -Ritz method was employed by Lee et al. [9] in studying the postbuckling response of FGM plates subject to compressive and thermal loads. Almeida et al. [10] extended the total Lagrange formulation proposed by Pacoste and Eriksson in [4] to investigate the geometrically nonlinear behaviour of FGM beams. A finite element formulation based on the linear exact shape functions was employed by Taeprasartsit [11] for buckling analysis of perfect and imperfect FGM columns. Recently, the first author and his coworkers [12–15] derived the corotational finite element formulations for large displacement analysis of FGM beam and frame structures. In these works, in order to prevent the formulations from the membrane locking, the average strain has been introduced to replace the membrane strain.

Sandwich structures with height strength-to-weight ratio are widely used in aerospace application such as skin of wings, aileron, and spoilers. In order to improve the performance of the structures in high thermal environment, FGMs could be incorporated in the sandwich construction. Several analyses, mainly the vibration and buckling, of functionally graded sandwich (FGSW) beams have been carried out in recent years [16–18]. To maintain minimum weight for a given mechanical loading condition, FGSW beams and frames are often designed to be slender and they might undergo large displacements during service. Investigation on the large displacement behaviour of FGSW beams and frames, to the authors’ best knowledge, has not been reported in the literature, and it is studied in this paper for the first time. To this end, a finite element formulation based on Euler-Bernoulli beam theory is derived in the context of the corotational approach and used in the investigation. Different from previous works, the interpolation functions for the displacement field are obtained by solving the equilibrium nonlinear equations of a beam segment. The exact interpolation functions lead to a balance between the axial and transverse displacements, and the finite element formulation based on these functions is free of the membrane locking [19]. The concept of the average membrane strain mentioned above is therefore not necessary to use in the present work. Using the derived formulation, the equilibrium equations are constructed and they are solved by an incremental-iterative procedure in combination with the arc-length control method. Numerical examples are given to show the accuracy of the derived formulation and to illustrate the effect of the material distribution and the core thickness ratio on the large displacement behaviour of the FGSW beam and frame structures.

#### 2. Finite Element Formulation

##### 2.1. FGSW Beam

Figure 1 shows a FGSW beam formed from a metallic soft core and two symmetric FGM skin layers in a Cartesian coordinate system. The -axis is chosen to be on the midplane of the beam and the -axis is directed upwards. Denoting the beam height and the soft core thickness by and , respectively, the FGM is assumed to be made of metal and ceramic with the volume fraction of ceramic, , varying in the thickness direction by the following power-law distribution:where is the nonnegative grading index. The volume fraction of metal is . As seen from (1), the top and bottom surfaces contain only ceramic, whereas the core is pure metal. The effective Young’s modulus of the beam, , evaluated by Voigt model readswhere and are Young’s moduli of ceramic and metal, respectively.