Advanced models for free vibration analysis of laminated beams with compact and thin-walled open/closed sections

In this paper, refined one-dimensional beam theories are implemented for the free vibration analysis of laminated beams with compact and thin-walled cross-sections. The proposed models are based on the Carrera Unified Formulation, which was formerly introduced for the analysis of plates and shells and recently extended to beam structures by the first author and his co-workers. Carrera Unified Formulation is a hierarchical modelling technique leading to very accurate and computationally efficient finite element theories. According to the latest developments in the framework of Carrera Unified Formulation, refined beam models are implemented using either Taylor-like or Lagrange-like polynomials in order to expand the unknown kinematic variables on the cross-section of the beam. Equivalent single layer models result from the former approach. On the other hand, if Lagrange polynomials are used, layer-wise models are produced. In this work, a classical one-dimensional finite element formulation along the beam length is used to develop numerical applications. A number of laminated beam structures are analyzed and particular attention is given to laminated box beams with open and closed cross-sections. The frequencies and the mode shapes obtained with the present refined beam elements are compared with solid/shell finite element solutions from the commercial code MSC/Nastran and, when possible, with those found in the literature. The modal assurance criterion is used for model-to-model comparisons so as to demonstrate the enhanced capabilities of the proposed formulation in investigating the free vibration characteristics of both compact and thin-walled box laminated beams.

[1]  Erasmo Carrera,et al.  Free Vibration Analysis of thin-walled cylinders reinforced with longitudinal and transversal stiffeners , 2013 .

[2]  J. R. Banerjee,et al.  Exact dynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structures , 2013 .

[3]  Erasmo Carrera,et al.  Variable Kinematic One-Dimensional Finite Elements for the Analysis of Rotors Made of Composite Materials , 2014 .

[4]  N. Pagano,et al.  Exact Solutions for Composite Laminates in Cylindrical Bending , 1969 .

[5]  P. Cunniff,et al.  The Vibration of Cantilever Beams of Fiber Reinforced Material , 1972 .

[6]  Erasmo Carrera,et al.  Analysis of Rotor Dynamic by One-Dimensional Variable Kinematic Theories , 2013 .

[7]  Mark V. Fulton,et al.  Free-Vibration Analysis of Composite Beams , 1991 .

[8]  Erasmo Carrera,et al.  Advanced beam formulations for free-vibration analysis of conventional and joined wings , 2012 .

[9]  Liviu Librescu,et al.  On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations , 2002 .

[10]  Tarun Kant,et al.  FREE VIBRATION ANALYSIS OF FIBER REINFORCED COMPOSITE BEAMS USING HIGHER ORDER THEORIES AND FINITE ELEMENT MODELLING , 1996 .

[11]  L. Gallimard,et al.  Composite beam finite element based on the Proper Generalized Decomposition , 2012 .

[12]  R. P. Shimpi,et al.  A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams , 2001 .

[13]  Erasmo Carrera,et al.  Refined One-Dimensional Formulations for Laminated Structure Analysis , 2012 .

[14]  K. Chandrashekhara,et al.  Free vibration of composite beams using a refined shear flexible beam element , 1992 .

[15]  Masoud Tahani,et al.  Analysis of laminated composite beams using layerwise displacement theories , 2007 .

[16]  Hidenori Murakami,et al.  Laminated Composite Plate Theory With Improved In-Plane Responses , 1986 .

[17]  Gaetano Giunta,et al.  Beam Structures: Classical and Advanced Theories , 2011 .

[18]  Erasmo Carrera,et al.  Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section , 2011 .

[19]  J. Reddy.  A Simple Higher-Order Theory for Laminated Composite Plates , 1984 .

[20]  J. N. Reddy,et al.  An exact solution for the bending of thin and thick cross-ply laminated beams , 1997 .

[21]  Erasmo Carrera,et al.  Refined beam elements with only displacement variables and plate/shell capabilities , 2012 .

[22]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[23]  Erian A. Armanios,et al.  The influence of coupling on the free vibration of anisotropic thin-walled closed-section beams , 1995 .

[24]  E. Carrera.  The effects of shear deformation and curvature on buckling and vibrations of cross-ply laminated composite shells , 1991 .

[25]  Olivier Polit,et al.  Assessment of the refined sinus model for the non-linear analysis of composite beams , 2009 .

[26]  Erasmo Carrera,et al.  Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies , 2012 .

[27]  Erasmo Carrera,et al.  Component-Wise Method Applied to Vibration of Wing Structures , 2013 .

[28]  Sung W. Lee,et al.  LARGE DEFLECTION STATIC AND DYNAMIC FINITE ELEMENT ANALYSES OF COMPOSITE BEAMS WITH ARBITRARY CROSS-SECTIONAL WARPING , 1989 .

[29]  Hassan Haddadpour,et al.  Flexural–torsional behavior of thin-walled composite beams with closed cross-section , 2007 .

[30]  Thuc P. Vo,et al.  Free vibration of thin-walled composite box beams , 2008 .

[31]  Erasmo Carrera,et al.  Refined 1D Finite Elements for the Analysis of Secondary, Primary, and Complete Civil Engineering Structures , 2015 .

[32]  Inderjit Chopra,et al.  Experimental-theoretical investigation of the vibration characteristics of rotating composite box beams , 1992 .

[33]  Erian A. Armanios,et al.  Free Vibration Analysis of Anisotropic Thin-Walled Closed-Section Beams , 1994 .

[34]  Yogesh M. Desai,et al.  Free vibrations of laminated beams using mixed theory , 2001 .

[35]  M. Seetharama Bhat,et al.  A new super convergent thin walled composite beam element for analysis of box beam structures , 2004 .

[36]  Taner Timarci,et al.  Free Vibration of Composite Box-Beams by Ansys , 2012 .

[37]  Mohamad S. Qatu,et al.  Vibrations of straight and curved composite beams: A review , 2013 .

[38]  J. R. Banerjee,et al.  Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions , 2014 .

[39]  E. Carrera.  Theories and finite elements for multilayered, anisotropic, composite plates and shells , 2002 .

[40]  Mohamad S. Qatu,et al.  Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions , 2012 .

[41]  Rakesh K. Kapania,et al.  Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation , 1989 .

[42]  Olivier Polit,et al.  A family of sinus finite elements for the analysis of rectangular laminated beams , 2008 .

[43]  S. Timoshenko,et al.  X. On the transverse vibrations of bars of uniform cross-section , 1922 .

[44]  Erasmo Carrera,et al.  Classical, refined and component-wise analysis of reinforced-shell structures , 2013 .

[45]  Li Jun,et al.  Dynamic stiffness analysis of laminated composite beams using trigonometric shear deformation theory , 2009 .

[46]  E. Carrera.  Historical review of Zig-Zag theories for multilayered plates and shells , 2003 .

[47]  E. Carrera,et al.  On the Effectiveness of Higher-Order Terms in Refined Beam Theories , 2011 .

[48]  P. Subramanian,et al.  Dynamic analysis of laminated composite beams using higher order theories and finite elements , 2006 .

[49]  Hiroyuki Matsunaga,et al.  Interlaminar stress analysis of laminated composite beams according to global higher-order deformation theories , 2002 .

[50]  Erasmo Carrera,et al.  Free vibration analysis of civil engineering structures by component-wise models , 2014 .

[51]  Dipak K. Maiti,et al.  A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates , 2013 .

[52]  Sergio Oller,et al.  Simple and accurate two-noded beam element for composite laminated beams using a refined zigzag theory , 2012 .

[53]  Mohamad S. Qatu,et al.  Theories and analyses of thin and moderately thick laminated composite curved beams , 1993 .

[54]  E. Carrera,et al.  Refined beam elements with arbitrary cross-section geometries , 2010 .

[55]  Erian A. Armanios,et al.  Influence of coupling on the free vibration of anisotropic thin-walled closed-section beams , 1995 .

[56]  Erasmo Carrera,et al.  Free vibration analysis of rotating composite blades via Carrera Unified Formulation , 2013 .

[57]  P. Vidal,et al.  A sine finite element using a zig-zag function for the analysis of laminated composite beams , 2011 .

[58]  Randall J. Allemang,et al.  A Correlation Coefficient for Modal Vector Analysis , 1982 .

[59]  Rakesh K. Kapania,et al.  Recent advances in analysis of laminated beams and plates. Part I - Sheareffects and buckling. , 1989 .

[60]  Karan S. Surana,et al.  Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites , 1990 .

[61]  Weiqiu Chen,et al.  Free vibration analysis of generally laminated beams via state-space-based differential quadrature , 2004 .

[62]  K. Chandrashekhara,et al.  Free vibration of composite beams including rotary inertia and shear deformation , 1990 .

[63]  K. Bathe.  Finite Element Procedures , 1995 .