A variational formulation for dynamic analysis of composite laminated beams based on a general higher-order shear deformation theory

Abstract This paper presents a general formulation for free and transient vibration analyses of composite laminated beams with arbitrary lay-ups and any boundary conditions. A modified variational principle combined with a multi-segment partitioning technique is employed to derive the formulation based on a general higher-order shear deformation theory. The material couplings of bending-stretching, bending-twist, and stretching-twist as well as the Poisson’s effect are taken into account. A considerable number of free and transient vibration solutions are presented for cross- and angle-ply laminated beams with various geometric and material parameters. Different combinations of free, simply-supported, pinned, clamped and elastic-supported boundary conditions are examined. The validity of the formulation is confirmed by comparing the present solutions with analytical and experimental results available in the literature and the ones obtained from finite element analyses. The accuracy of several higher-order shear deformable beam theories for predicting the vibrations of laminated beams has been ascertained. Results of parametric studies for composite beams with different orthotropic ratios, fiber orientations, layer numbers and boundary conditions are also discussed. The present formulation is versatile in the sense that it is capable of accommodating a variety of beam theories available in the literature, and allows the use of different polynomials as admissible functions for composite beams, such as the Chebyshev and Legendre orthogonal polynomials, and the ordinary power polynomials. Moreover, it permits to deal with the linear vibration problems for thin and thick beams subjected to dynamic loads and boundary conditions of arbitrary type.

[1]  C. Bert,et al.  The behavior of structures composed of composite materials , 1986 .

[2]  Dimitris A. Saravanos,et al.  High-order layerwise mechanics and finite element for the damped dynamic characteristics of sandwich composite beams , 2004 .

[3]  Kostas P. Soldatos,et al.  A transverse shear deformation theory for homogeneous monoclinic plates , 1992 .

[4]  Luciano Demasi,et al.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations , 2009 .

[5]  Tarun Kant,et al.  Transient dynamics of laminated beams: an evaluation with a higher-order refined theory , 1998 .

[6]  António J.M. Ferreira,et al.  Thick Composite Beam Analysis Using a Global Meshless Approximation Based on Radial Basis Functions , 2003 .

[7]  Yeoshua Frostig,et al.  Boundary Condition Effects in Free Vibrations of Higher-Order Soft Sandwich Beams , 2002 .

[8]  K. Chandrashekhara,et al.  Some observations on the modeling of laminated composite beams with general lay-ups , 1991 .

[9]  M. Levinson,et al.  A new rectangular beam theory , 1981 .

[10]  R. P. Shimpi,et al.  A Review of Refined Shear Deformation Theories for Isotropic and Anisotropic Laminated Beams , 2001 .

[11]  Sébastien Mistou,et al.  Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity , 2003 .

[12]  Prediction of the dynamic response of composite sandwich beams under shock loading , 2010 .

[13]  Abdolreza Ohadi,et al.  A numerical study of free and forced vibration of composite sandwich beam with viscoelastic core , 2010 .

[14]  Vebil Yıldırım,et al.  Investigation of the rotary inertia and shear deformation effects on the out-of-plane bending and torsional natural frequencies of laminated beams , 2000 .

[15]  S.M.R. Khalili,et al.  High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method , 2012 .

[16]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation.: Part II: Layerwise theories , 2009 .

[17]  A. R. Fiouz,et al.  Dynamic investigation of laminated composite beams with shear and rotary inertia effect subjected to the moving oscillators using FEM , 2011 .

[18]  A. H. Shah,et al.  NATURAL VIBRATIONS OF LAMINATED COMPOSITE BEAMS BY USING MIXED FINITE ELEMENT MODELLING , 2002 .

[19]  Hiroyuki Matsunaga,et al.  VIBRATION AND BUCKLING OF MULTILAYERED COMPOSITE BEAMS ACCORDING TO HIGHER ORDER DEFORMATION THEORIES , 2001 .

[20]  Faruk Firat Calim,et al.  Free and forced vibrations of non-uniform composite beams , 2009 .

[21]  Guang Meng,et al.  Free and forced vibration analysis of uniform and stepped circular cylindrical shells using a domain decomposition method , 2013 .

[22]  Guang Meng,et al.  A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia , 2013 .

[23]  S. Kapuria,et al.  Efficient layerwise finite element model for dynamic analysis of laminated piezoelectric beams , 2006 .

[24]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[25]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part IV: Zig-zag theories , 2009 .

[26]  Huu-Tai Thai,et al.  Free vibration of axially loaded rectangular composite beams using refined shear deformation theory , 2012 .

[27]  Dynamic response of antisymmetric cross-ply laminated composite beams with arbitrary boundary conditions , 1996 .

[28]  Krishna Av Murty.  Toward a consistent beam theory , 1984 .

[29]  L. Demasi.  Treatment of stress variables in advanced multilayered plate elements based upon Reissner’s mixed variational theorem , 2006 .

[30]  Steven Nutt,et al.  Consistent Higher-Order Dynamic Equations for Soft-Core Sandwich Beams , 2004 .

[31]  B. Omidvar.  Shear Coefficient in Orthotropic Thin-Walled Composite Beams , 1998 .

[32]  Dimitris A. Saravanos,et al.  Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors , 2007 .

[33]  E. Reissner,et al.  On transverse bending of plates, including the effect of transverse shear deformation☆ , 1975 .

[34]  Shen Rongying,et al.  Dynamic finite element method for generally laminated composite beams , 2008 .

[35]  C. Soares,et al.  Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory , 2012 .

[36]  M. Topçu,et al.  Free Vibration Analysis of the Laminated Composite Beams by Using DQM , 2009 .

[37]  A. Ellakany.  Calculation of higher natural frequencies of simply supported elastic composite beams using Riccati matrix method , 2008 .

[38]  C. Guedes Soares,et al.  A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates , 2012 .

[40]  Guang Meng,et al.  A modified variational approach for vibration analysis of ring-stiffened conical–cylindrical shell combinations , 2013 .

[41]  J. Berthelot,et al.  Damping analysis of laminated beams and plates using the Ritz method , 2006 .

[42]  M. Touratier,et al.  An efficient standard plate theory , 1991 .

[43]  Metin Aydogdu,et al.  A new shear deformation theory for laminated composite plates , 2009 .

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

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

[46]  N. E. Meiche,et al.  A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate , 2011 .

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

[48]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part III: Advanced mixed high order shear deformation theories , 2009 .

[49]  Gaetano Giunta,et al.  Free vibration analysis of composite beams via refined theories , 2013 .

[50]  Vladimír Panc,et al.  Theories of elastic plates , 1975 .

[51]  I. Ritchie,et al.  The dynamic elastic behaviour of a fibre-reinforced composite sheet. I. The precise experimental determination of the principal elastic moduli , 1975 .

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

[53]  M. Aydogdu.  Free Vibration Analysis of Angle-ply Laminated Beams with General Boundary Conditions , 2006 .

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

[55]  Metin Aydogdu,et al.  Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method , 2005 .

[56]  Chien Wei-zang.  Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals , 1983 .

[57]  Zeki Kiral,et al.  Damped Response of Symmetric Laminated Composite Beams to Moving Load with Different Boundary Conditions , 2009 .

[58]  C. Soares,et al.  A new higher order shear deformation theory for sandwich and composite laminated plates , 2012 .

[59]  A. M. Ellakany,et al.  A combined transfer matrix and analogue beam method for free vibration analysis of composite beams , 2004 .

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

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

[62]  Frederic Ward Williams,et al.  EXACT DYNAMIC STIFFNESS MATRIX FOR COMPOSITE TIMOSHENKO BEAMS WITH APPLICATIONS , 1996 .

[64]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[65]  T. Jiang,et al.  STRESS DISTRIBUTION AND DEFORMATION OF ADHESIVE-BONDED LAMINATED COMPOSITE BEAMS , 1989 .

[66]  J. N. Reddy,et al.  Free vibration of cross-ply laminated beams with arbitrary boundary conditions , 1994 .

[67]  Luciano Demasi,et al.  Quasi-3D analysis of free vibration of anisotropic plates , 2006 .

[68]  Alan Jeffrey,et al.  Handbook of mathematical formulas and integrals , 1995 .

[69]  K. Chandrashekhara,et al.  Analytical solutions to vibration of generally layered composite beams , 1992 .

[70]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part V: Results , 2009 .

[71]  E. Carrera.  Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells , 2001 .

[72]  Tarun Kant,et al.  Analytical solution to the dynamic analysis of laminated beams using higher order refined theory , 1997 .

[73]  M. Stein,et al.  Nonlinear theory for plates and shells including the effects of transverse shearing , 1986 .

[74]  C. Guedes Soares,et al.  Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates , 2012 .

[75]  Maurice Touratier,et al.  A refined theory of laminated shallow shells , 1992 .

[76]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations , 2002 .

[77]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices , 2002 .

[78]  Moshe Eisenberger,et al.  Dynamic stiffness analysis of laminated beams using a first order shear deformation theory , 1995 .

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

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

[81]  N. Ganesan,et al.  Static analysis of functionally graded beams using higher order shear deformation theory , 2008 .

[82]  A. H. Tanrikulu,et al.  Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories , 2008 .

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

[84]  Guang Meng,et al.  A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries , 2013 .

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

[86]  M. Kargarnovin,et al.  An analytical approach for the free vibration analysis of generally laminated composite beams with shear effect and rotary inertia , 2012 .

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

[88]  C. Soares,et al.  Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory , 2011 .