Abstract
The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.
Similar content being viewed by others
References
Kapania RK, Raciti S (1989) Recent advances in analysis of laminated beams and plates: Part I. Shear effects and buckling; Part II. Vibrations and wave propagation. AIAA J 27:923–946
Abramovich H (1994) Thermal buckling of cross-ply composite laminates using first-order shear deformation theory. Compos Struct 28:201–213
Eisenberger M, Abramovich H, Shulepov O (1995) Dynamic stiffness analysis of laminated beams using first order shear deformation theory. Compos Struct 31:265–271
Teboub Y, Hajela P (1995) Free vibration of generally layered composite beams using symbolic computations. Compos Struct 33:123–134
Abramovich H, Eisenberger M, Shulepov O (1996) Vibrations and buckling of cross-ply nonsymmetric laminated composite beams. AIAA J 34:1064–1069
Banerjee JR (1998) Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Comput Struct 69:197–208
Mahapatra DR, Gopalakrishnan S (2003) A spectral finite element model for analysis of axial-flexural-shear coupled wave propagation in laminated composite beams. Compos Struct 59:67–88
Ruotolo R (2004) A spectral element for laminated composite beams: theory and application to pyroshock analysis. J Sound Vib 270:149–169
Ellakany AM (2008) Calculation of higher natural frequencies of simply supported elastic composite beams using Riccati matrix method. Meccanica 43:523–532
Bhimaraddi A, Chandrashekhara K (1991) Some observations on the modeling of laminated composite beams with general lay-ups. Compos Struct 19:371–380
Soldatos KP, Elishakoff I (1992) A transverse shear and normal deformable orthotropic beam theory. J Sound Vib 154:528–533
Chandrashekhara K, Bangera KM (1992) Free vibration of composite beams using a refined shear flexible beam element. Comput Struct 43:719–727
Singh MP, Abdelnaser AS (1992) Random response of symmetric cross-ply composite beams with arbitrary boundary conditions. AIAA J 30:1081–1088
Savoia M, Laudiero F, Tralli A (1993) A refined theory for laminated beams: part I—a new high order approach. Meccanica 28:39–51
Savoia M, Tralli A, Laudiero F (1993) A refined theory for laminated beams: part II—an iterative variational approach. Meccanica 28:217–225
Khdeir AA, Reddy JN (1994) Free vibration of cross-ply laminated beams with arbitrary boundary conditions. Int J Eng Sci 32:1971–1980
Khdeir AA, Reddy JN (1997) Buckling of cross-ply laminated beams with arbitrary boundary conditions. Compos Struct 37:1–3
Song SJ, Waas AM (1997) Effects of shear deformation on buckling and free vibration of laminated composite beams. Compos Struct 37:33–43
Karama M, Abou Harb B, Mistou S, Caperaa S (1998) Bending, buckling and free vibration of laminated composite with a transverse shear stress continuity model. Composites, Part B 29:223–234
Shi G, Lam KY (1999) Finite element vibration analysis of composite beams based on higher-order beam theory. J Sound Vib 219:707–721
Ghugal YM, Shimpi RP (2001) A review of refined shear deformation theories for isotropic and anisotropic laminated beams. J Reinf Plast Compos 20:255–272
Matsunaga H (2001) Vibration and buckling of multilayered composite beams according to higher order deformation theories. J Sound Vib 246:47–62
Khdeir AA (2001) Thermal buckling of cross-ply laminated composite beams. Acta Mech 149:201–213
Kapuria S, Dumir PC, Jain NK (2004) Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Compos Struct 64:317–327
Aydogdu M (2005) Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Int J Mech Sci 47:1740–1755
Aydogdu M (2006) Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Compos Sci Technol 66:1248–1255
Aydogdu M (2007) Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions. Compos Sci Technol 67:1096–1104
Leung AYT (1993) Dynamic stiffness and substructures. Springer, London
Doyle JF (1997) Wave propagation in structures. Springer, New York
Lee U (2004) Spectral element method in structural dynamics. Inha University Press, Incheon
Gopalakrishnan S, Chakraborty A, Mahapatra DR (2008) Spectral finite element method. Springer, London
Jones RM (1976) Mechanics of composite materials. McGraw-Hill, New York
Wolfram S (1991) Mathematica: a system for doing mathematics by computer, 2nd edn. Addison-Wesley, Reading
Wittrick WH, Williams FW (1971) A general algorithm for computing natural frequencies of elastic structures. Q J Mech Appl Math 24:263–284
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jun, L., Hongxing, H. Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory. Meccanica 46, 1299–1317 (2011). https://doi.org/10.1007/s11012-010-9388-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-010-9388-7