A Ritz Approximation to the Deformation of Anisotropic Slender Structures with Finite-Size Cross Sections

A one-dimensional theory of slender structures with heterogeneous anisotropic materials is presented. It expands Cosserat’s description of beam kinematics by allowing deformation of the beam cross sections. For that purpose, a Ritz approximation is introduced on the cross-sectional warping field, which defines additional elastic degrees of freedom (finitesection modes) in the 1-D model. This results in an extended set of beam dynamic equations that includes direct measures of both the large global displacement and rotations of a certain reference line, and the small local deformations of the cross sections. Two situations of interest are then studied in which this approach provides a simpler alternative to nonlinear shell models: First, we look at the detailed structural response of thin-walled composite beams with distributed loads. In particular, the case of a composite construction with embedded piezoelectric actuators is considered. Second, this methodology is applied to study the low-frequency response characterization of a thin-walled composite beam. Numerical results are presented in both cases, in which a reduced set of finite-section modes allows a full characterization of the actual 3-D structure within a strictly 1-D framework solution.

[1]  Dewey H. Hodges,et al.  Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor , 1987 .

[2]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[3]  Carlos E. S. Cesnik,et al.  Finite element solution of nonlinear intrinsic equations for curved composite beams , 1995 .

[4]  J. B. Kosmatka,et al.  Extension-bend-twist coupling behavior of nonhomogeneous anisotropic beams with initial twist , 1992 .

[5]  Patrice Cartraud,et al.  Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method , 2001 .

[6]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[7]  G. E. O. Widera,et al.  REFINED ENGINEERING BEAM THEORY BASED ON THE ASYMPTOTIC EXPANSION APPROACH , 1991 .

[8]  John B. Kosmatka,et al.  Torsion and flexure of a prismatic isotropic beam using the boundary element method , 2000 .

[9]  Dewey H. Hodges,et al.  Nonlinear Composite Beam Theory , 2006 .

[10]  D. F. Parker.  An asymptotic analysis of large deflections and rotations of elastic rods , 1979 .

[11]  Khanh Chau Le,et al.  Vibrations of Shells and Rods , 1999 .

[12]  Carlos E. S. Cesnik,et al.  Cross-sectional analysis of nonhomogeneous anisotropic active slender structures , 2005 .

[13]  Carlos E. S. Cesnik,et al.  Modeling of High Aspect Ratio Active Flexible Wings for Roll Control , 2002 .

[14]  Inderjit Chopra,et al.  Assessment of Composite Rotor Blade Modeling Techniques , 1999 .

[15]  D. Hodges.  A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams , 1990 .

[16]  Dewey H. Hodges,et al.  Theory of Anisotropic Thin-Walled Beams , 2000 .

[17]  Inderjit Chopra,et al.  Review of State of Art of Smart Structures and Integrated Systems , 2002 .

[18]  Carlos E. S. Cesnik,et al.  On the one-dimensional modeling of camber bending deformations in active anisotropic slender structures , 2008 .

[19]  Dewey H. Hodges,et al.  Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams , 2004 .

[20]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[21]  V. Berdichevskiĭ.  On the energy of an elastic rod , 1981 .

[22]  Carlos E. S. Cesnik,et al.  Assessment of Beam Modeling Methods for Rotor Blade Applications , 2001 .

[23]  Carlos E. S. Cesnik,et al.  VABS: A New Concept for Composite Rotor Blade Cross-Sectional Modeling , 1995 .

[24]  Dewey H. Hodges,et al.  A Beam Theory for Large Global Rotation, Moderate Local Rotation, and Small Strain , 1988 .

[25]  M. Borri,et al.  Anisotropic beam theory and applications , 1983 .

[26]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[27]  Werner Wagner,et al.  Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections , 2001 .

[28]  Carlos E. S. Cesnik,et al.  Refined theory of composite beams: The role of short-wavelength extrapolation , 1996 .