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Research Papers

# Consistent Asymptotic Expansion Multiscale Formulation for Heterogeneous Column Structure

[+] Author and Article Information
Dongdong Wang1

Department of Civil Engineering,  Xiamen University, Xiamen, Fujian 361005, Chinaddwang@xmu.edu.cn

Pinkang Xie

Department of Civil Engineering,  Xiamen University, Xiamen, Fujian 361005, Chinapkxie@qq.com

Lingming Fang

Department of Civil Engineering,  Xiamen University, Xiamen, Fujian 361005, Chinazpflm@hotmail.com

1

Corresponding author.

J. Eng. Mater. Technol 134(3), 031006 (May 07, 2012) (7 pages) doi:10.1115/1.4006505 History: Received September 26, 2011; Revised February 09, 2012; Published May 04, 2012; Online May 07, 2012

## Abstract

A consistent asymptotic expansion multiscale formulation is presented for analysis of the heterogeneous column structure, which has three dimensional periodic reinforcements along the axial direction. The proposed formulation is based upon a new asymptotic expansion of the displacement field. This new multiscale displacement expansion has a three dimensional form, more specifically, it takes into account the axial periodic property but simultaneously keeps the cross section dimensions in the global scale. Thus, this formulation inherently reflects the characteristics of the column structure, i.e., the traction free condition on the circumferential surfaces. Subsequently, the global equilibrium problem and the local unit cell problem are consistently derived based upon the proposed asymptotic displacement field. It turns out that the global homogenized problem is the standard axial equilibrium equation, while the local unit cell problem is completely three dimensional which is subjected to the periodic boundary condition on axial surfaces as well as the traction free condition on circumferential surfaces of the unit cell. Thereafter, the variational formulation and finite element discretization of the unit cell problem are discussed. The effectiveness of the present formulation is illustrated by several numerical examples.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures

Figure 1

A column structure with three dimensional heterogeneities: (a) global structure; (b) local unit cells

Figure 2

Dimensional notations for the unit cell

Figure 3

Finite element mesh for unit cell

Figure 4

Microscopic deformation for homogenous unit cell: (a) local fluctuating deformation; (b) total deformation

Figure 5

Microscopic deformation for heterogeneous unit cell with c = 0.04 m: (a) local fluctuating deformation; (b) total deformation

Figure 6

Microscopic deformation for heterogeneous unit cell with c = 0.03 m: (a) local fluctuating deformation; (b) total deformation

Figure 7

Microscopic deformation for heterogeneous unit cell with c = 0.02 m: (a) local fluctuating deformation; (b) total deformation

Figure 8

Microscopic deformation for heterogeneous unit cell with EI2=10GPa: (a) local fluctuating deformation; (b) total deformation

Figure 9

A column under self-weight and external force: (a) problem statement; (b) mesh for direct finite element simulation

Figure 10

Microscopic axial stress distribution at PM by multiscale analysis: (a) overall distribution; (b) z-cross section distribution; (c) y-cross section distribution

Figure 11

Microscopic axial stress distribution at PM by direct finite element simulation: (a) overall distribution; (b) z-cross section distribution; (c) y-cross section distribution

Figure 12

Microscopic axial stress distribution at PB by multiscale analysis: (a) overall distribution; (b) z-cross section distribution; (c) y-cross section distribution

Figure 13

Microscopic axial stress distribution at PB by direct finite element simulation: (a) overall distribution; (b) z-cross section distribution; (c) y-cross section distribution

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