Nonlinear Analysis of Plane Frames Using a Co-Rotating Timoshenko Beam Element
Journal: Journal of Building Technology DOI: 10.32629/jbt.v6i2.2552
Abstract
The present work describes a co-rotating shear flexible beam element without shear locking and integrating Euler-Bernoulli's and Timoshenko's beam theories. The co-rotational kinematics is based on the separation of the motion in deformational and rigid body components. The deformation of the beam element is composed by three natural modes of deformation: the extension mode, the symmetric bending mode, and the anti-symmetric bending mode. The respective generalized stresses from these natural modes are self-balanced, allowing the achievement of a consistent tangent stiffness matrix. In this paper, it is detailed and deduced all the algebraic steps for the deduction of the elastic stiffness matrix, the geometric stiffness matrix, and the co-rotation stiffness matrix. Some examples are presented and the numerical results demonstrate that the beam element here presented is able to handle large rotations.
Keywords
Bernoulli/Timoshenko beam element; corotational kinematic; deformation's natural modes
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[3] C.A. Felippa, B. Haugen. 2005. A unified formulation of small-strain corotational finite elements: I. Theory. Computer Methods in Applied Mechanics and Engineering, 81: 131-150.
[4] M. Mostafa, M.V. Sivaselvan, C.A. Felippa. 2013. A solid-shell corotational element based on ANDES, ANS and EAS for geometrically nonlinear structural analysis. International Journal for Numerical Methods in Engineering, 95: 145-180.
[5] W.T. Matias e, L.M. Bezerra. 2009. Uma abordagem unificada da formulacao corotacional para elementos de trelica 2D, trelica 3D e viga 2D. Revista Internacional de Métodos Numéricos para Cálculo y Diseno en Ingeniería, 25(2): 163-190.
[6] T.N. Le, J.M. Battini, M. Hjiaj. 2011. Efficient formulation for dynamics of corotational 2D beams. Computational Mechanics, 48(2): 153-161.
[7] T.N. Le, J.M. Battini, M. Hjiaj. 2014. A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures. Computer Methods in Applied Mechanics and Engineering, 269: 538-565.
[8] J. Argyris, H. Balmer, J.St. Doltsinis, et al. 1979. Finite element method - the natural approach. Computer Methods in Applied Mechanics and Engineering, 17/18: 1-106.
[9] J. Argyris, H.O. Hilpert, G.A. Malejannakis, D.W. Scharpf. 1979. On the geometrical stiffness of a beam in space - A consistent V.W. approach. Computer Methods in Applied Mechanics and Engineering, 20: 105-131.
[10] S. Krenk. 1994. A general format for curved and nonhomogeneous beam elements. Computers & Structures, 50: 449-454.
[11] S. Krenk. 2009. Non-Linear Modeling and Analysis of Solids and Structures, Cambridge University Press.
[12] M.J. Turner, R.W. Clough, H.C. Martin, L.J. Topp. 1956. Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences, 23: 805-824.
[13] J.S. Przemieniecki. 1968. Theory of Matrix Structural Analysis, McGraw-Hill.
[14] H.C. Martin. 1966. On the derivation of stiffness matrices for the analysis of large deflection and stability problems, AFFDL-TR-66-80. Air Force Institute of Technology, pp. 697-716.
[15] S.L. Chan, P.P.T. Chui. 2000. Non-linear static and cyclic analysis of steel frames with semi-rigid connections, Elsevier, Amsterdan.
[16] W.T. Matias. 2002. El control variable de los desplazamientos en el análisis no linealelástico de estructuras de barras. Revista Internacional de Métodos Numéricospara Cálculo y Diseno en Ingeniería, 18(4): 549-572.
[17] F. Fujii, K.K. Choong, S.X. Gong. 1992. Variable displacement control to overcome turnning points of nonlinear elastic frames. Computers & Structures, 44(1/2): 133-136.
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