Nonlinear Static Analysis of Laminated Composite Hollow Beams with Super-Elliptic Cross-Sections
In this paper geometrically nonlinear static behavior of laminated composite hollow super-elliptic beams is investigated using generalized differential quadrature method. Super-elliptic beam can have both oval and elliptic cross-sections by adjusting parameters in super-ellipse formulation (also known as Lamé curves). Equilibrium equations of super-elliptic beam are obtained using the virtual work principle. Geometric nonlinearity is taken into account using von-Kármán nonlinear strain-displacement relations. Spatial derivatives in strains are expressed with the generalized differential quadrature method. Transverse shear effect is considered through the first-order shear deformation theory. Static equilibrium equations are solved using Newton-Raphson method. Several composite super-elliptic beam problems are solved with the proposed method. Effects of layer orientations of composite material, boundary conditions, ovality and ellipticity on bending behavior are investigated.
 L. Guo, S. Yang, H. Jiao, “Behavior of thin-walled circular hollow section tubes subjected to bending,” Thin-Walled Structures, vol. 73, pp. 281-289, 2013.
 D. Karagiozova, T. X. Yu, G. Lu, X. Xiang, “Response of a circular metallic hollow beam to an impulsive loading,” Thin-Walled Structures, vol. 80, pp. 80-90, 2014.
 D. Karagiozova, T. X. Yu, G. Lu, “Transverse blast loading of hollow beams with square cross-sections,” Thin-Walled Structures, vol. 62, pp. 169-178, 2013.
 B. Zheng, G. Shu, L. Xin, R. Yang, Q. Jiang, “Study on the Bending Capacity of Cold-formed Stainless Steel Hollow Sections,” Structures, vol. 8, pp. 63-74, 2016.
 Z. Li, D. Yang, “Thermal post buckling analysis of anisotropic laminated beams with tubular cross-section based on higher-order theory,” Ocean Engineering, vol. 115, pp. 93-106, 2016.
 H. Asadi, M. M. Aghdam, “Large amplitude vibration and post-buckling analysis of variable cross-section composite beams on nonlinear elastic foundation,” International Journal of Mechanical Sciences, vol. 79, pp. 47-55, 2014.
 Y. Zhao, Y. Huang, M. Guo, “A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory” Composite Structures, vol. 168, pp. 277-284, 2017.
 E. Ghafari, J. Rezaeepazhand, “Isogeometric analysis of composite beams with arbitrary cross-section using dimensional reduction method” Computer Methods in Applied Mechanics and Engineering, vol. 318, pp. 594-618, 2017.
 P. Jiao, W. Borchani, H. Hasni, A. Alavi, N. Lajnef, “Post-buckling response of non-uniform cross-section bilaterally constrained beams” Mechanics Research Communications, vol. 78, pp. 42-50, 2016.
 K. H. Law, L. Gardner, “Lateral instability of elliptical hollow section beams” Engineering Structures, vol. 37, pp. 152-166, 2012.
 J. Murin, V. Goga, M. Aminbaghai, J. Hrabovsky, T. Sedlar, H. A. Mang, “Measurement and modelling of torsional warping free vibrations of beams with rectangular hollow cross-sections” Engineering Structures, vol. 136, pp. 68-76, 2017.
 J. N. Reddy, “Geometrically Nonlinear Transient Analysis of Laminated Composite Plates” AIAA Journal, vol. 21(4), pp. 621-628, 1983.
 G. R. Cowper, “The Shear Coefficient in Timoshenko's Beam Theory” Journal of Applied Mechanics, vol. 33(2), pp. 335-340, 1966.
 M, Hajianmaleki, M. S. Qatu. “Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions” Composite Part B: Eng, vol. 43, pp. 1767–75, 2012.
 G. She, F. Yuan, Y. Ren, “Nonlinear analysis of bending, thermal buckling and post-buckling for functionally graded tubes by using a refined beam theory” Composite Structures, vol. 165, pp. 74-82, 2017.