Journal of Mechanical Science and Technology

, Volume 26, Issue 1, pp 161–172

# A cross-sectional analysis of composite beams based on asymptotic framework

• Joonho Jeong
• Jun-Sik Kim
• Yeon June Kang
• Maenghyo Cho
Article

## Abstract

This paper presents the accurate prediction of static behavior of composite beams with arbitrary cross-sections. The asymptotic recursive formulation is reviewed first, where the initial three-dimensional problems are split into the macroscopic 1D problems and the microscopic 2D problems. The finite element formulation for the microscopic 2D problems is then presented in order to find the crosssectional warping solutions. The warping solutions obtained contribute the cross-sectional properties to the macroscopic 1D problems. The end effect of the 1D beam problem is also considered via the kinematic correction for a displacement prescribed boundary. The approach presented is applied to the beams with relatively complicated material distributions and cross-sectional geometry. As numerical test-beds, a three-layered sandwich beam and a composite beam with the multi-cell cross-section are taken to analyze the local deformation. A parametric study is also carried out to investigate the significance of shear deformation due to the cross-sectional orthotropic characteristics. The cross-sectional deformation is predicted based on the asymptotic framework. The accuracy of the present approach is assessed by comparing the results obtained with the 3D FEM solutions obtained by ANSYS.

### Keywords

Asymptotic expansion method Cross-sectional analysis Warping solution Composite beams

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### References

1. [1]
J.-S. Kim and K. W. Wang, Vibration analysis of composite beams with end effects via the formal asymptotic method, ASME: Journal of Vibration and Acoustics, 132 (2010) 041003:1–8.Google Scholar
2. [2]
V. L. Berdichevsky, On the energy of an elastic rod, Journal of Applied Mathematics and Mechanics (PMM), 45(4) (1981) 518–529.
3. [3]
C. E. S. Cesnik, V. G. Sutyrin and D. H. Hodges, Refined theory of composite beams: the role of short-wavelength extrapolation, International Journal of Solids and Structures, 33(10) (1996) 1387–1408.
4. [4]
C. E. S. Cesnik, V. G. Sutyrin and D. H. Hodges, VABS: a new concept for composite rotor blade cross-sectional modeling, Journal of American Helicopter Society, 42(1) (1997) 27–38.
5. [5]
B. Popescu and D. H. Hodges, On asymptotically correct Timoshenko-like anisotropic beam theory, International Journal of Solids and Structures, 37 (2000) 535–558.
6. [6]
W. Yu, D. H. Hodges, V. V. Volovoi and C. E. S. Cesnik, On Timoshenko-like modeling of initially curved and twisted composite beams, International Journal of Solids and Structures 39 (2002) 5101–5121.
7. [7]
L. Trabucho and J. M. Viano, Mathematical modeling of rods. In: Ciarlet PG, Lions JL. Handbook of Numerical Analysis, North-Holland (1996) Vol. 4.Google Scholar
8. [8]
R. D. Gregory and F. Y. M. Wan, Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory, Journal of Elasticity, 14 (1984) 27–64.
9. [9]
N. Buannic and P. Cartraud, Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method, International Journal of Solids and Structures, 38 (2001a) 7139–7161.
10. [10]
N. Buannic, P. Cartraud, Higher-order effective modeling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution, International Journal of Solids and Structures, 38 (2001b) 7168–7180.Google Scholar
11. [11]
H. Fan and G. E. O. Widera, On the proper boundary conditions for a beam, Journal of Applied Mechanics, 59 (1992) 915–922.
12. [12]
H. Fan and G. E. O. Widera, On use of variational principles to derive beam boundary conditions, Journal of Applied Mechanics, 61 (1992) 470–471.
13. [13]
C. O. Horgan and J. G. Simmonds, Asymptotoic analysis of an end-loaded transversely isotropic, elastic, semi-infinite strip weak in shear, International Journal of Solids and Structures, 27 (1991) 1895–1914.
14. [14]
J. M. Duva and J. G. Simmonds, The usefulness of elementary theory for the linear vibrations of layered, orthotropic elastic beams and corrections due to two-dimensional end effect, Journal of Applied Mechanics, 58 (1991) 175–180.
15. [15]
J.-S. Kim, M. Cho and E. C. Edward, An asymptotic analysis of composite beams with kinematically corrected end effects, International Journal of Solids and Structures, 45 (2008) 1954–1977.
16. [16]
M. Peters and K. Hackl, Numerical aspects of the eXtended finite element method, Proceeding in Applied Mathematics and Mechanics, 5 (2005), 355–356.
17. [17]
R. L. Taylor, P. J. Beresford and E. L. Wilson, A nonconforming element for stress analysis, International Journal for Numerical Methods in Engineering, 10 (1976) 1211–1219.
18. [18]
ANSYS, ANSYS user’s guide release 9.0, 2004.Google Scholar

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

• Joonho Jeong
• 1
• Jun-Sik Kim
• 2
• Yeon June Kang
• 3
• Maenghyo Cho
• 4
1. 1.Interdisciplinary Program in Automotive EngineeringSeoul National UniversitySeoulKorea
2. 2.Department of Intelligent Mechanical EngineeringKumoh National Institute of TechnologyGumi, GyeongbukKorea
3. 3.School of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulKorea
4. 4.WCU Multiscale Mechanical Design Division, School of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulKorea