FE Modeling and Analysis of Isotropic and Orthotropic Beams Using First Order Shear Deformation Theory

Author(s)
M. Adnan Elshafei

ABSTRACT

In the present work, a finite element model is developed to analyze the response of isotropic and orthotropic beams, a common structural element for aeronautics and astronautic applications. The assumed field displacements equations of the beams are represented by a first order shear deformation theory, the Timoshenko beam theory. The equations of motion of the beams are derived using Hamilton’s principle. The shear correction factor is used to improve the obtained results. A MATLAB code is constructed to compute the natural frequencies and the static deformations for both types of beams with different boundary conditions. Numerical calculations are carried out to clarify the effects of the thickness-to-length ratio on both the Eigen values and the deflections of the beams due to the applied mechanical load. The obtained results of the proposed model are compared to the available results of other investigators, good agreement is generally obtained.

Cite this paper

M. Elshafei, "FE Modeling and Analysis of Isotropic and Orthotropic Beams Using First Order Shear Deformation Theory,"*Materials Sciences and Applications*, Vol. 4 No. 1, 2013, pp. 77-102. doi: 10.4236/msa.2013.41010.

M. Elshafei, "FE Modeling and Analysis of Isotropic and Orthotropic Beams Using First Order Shear Deformation Theory,"

References

[1] F. G. Yuan and R. E. Miller, “A New Finite Element for Laminated Composite Beams,” Computers and Structures, Vol. 31, No. 5, 1989, pp. 737-745. doi:10.1016/0045-7949(89)90207-1

[2] K. Chandrashekhara and K. M. Bangera, “Free Vibration of Composite Beams Using a Refined Shear Flexible Beam Element,” Computers and Structures, Vol. 43, No. 4, 1992, pp. 719-727. doi:10.1016/0045-7949(92)90514-Z

[3] Z. Friedman and J. B. Kosmatka, “An Improved Two Node Timoshenko Beam Finite Element,” Computer and Structures, Vol. 47, No. 3, 1993, pp. 473-481. doi:10.1016/0045-7949(93)90243-7

[4] T. Lidstrom, “An Analytical Energy Expression for Equilibrium Analysis of 3-D Timoshenko Beam Element,” Computers and Structures, Vol. 52, No. 1, 1994, pp. 95 101. doi:10.1016/0045-7949(94)90259-3

[5] S. R. Bhate, U. N. Nayak and A. V. Patki, “Deformation of Composite Beam Using Refined Theory,” Computers and Structures, Vol. 54, No. 3, 1995, pp. 541-546. doi:10.1016/0045-7949(94)00354-6

[6] S. M. Nabi and N. Ganesan, “Generalized Element for the Free Vibration Analysis of Composite Beams,” Computers and Structures, Vol. 51, No.5, 1994, pp. 607-610. doi:10.1016/0045-7949(94)90068-X

[7] E. A. Armanios and A. M. Badir, “Free Vibration Analysis of the Anisotropic Thin-Walled Closed-Section Beams,” Journal of the American Institute of Aeronautics and Astronautics, Vol. 33, No. 10, 1995, pp. 1905-1910. doi:10.2514/3.12744

[8] V. Giavotto, M. Borri, P. Mantegazza., G. Ghiringhelli, V. Carmash, G. Maffioli and F. Mussi, “Anisotropic Beam Theory And Applications,” Computers and Structures, Vol. 16, No. 4, 1983, pp. 403-413. doi:10.1016/0045-7949(83)90179-7

[9] D. Hagodes, A. Atilgan, M. Fulton and L. Rehfield, “Free Vibration Analysis of Composite Beams,” Journal of the American Helicopter Society, Vol. 36, No. 3, 1991, pp. 36-47. doi:10.4050/JAHS.36.36

[10] R. Chandra and I. Chopra, “Experimental and Theoretical Investigation of the Vibration Characteristics of Rotating Composite Box Beams,” Journal of Aircraft, Vol. 29, No. 4, 1992, pp. 657-664. doi:10.2514/3.46216

[11] S. R. Rao and N. Ganesan, “Dynamic Response of Tapered Composites Beams Using Higher Order Shear Deformation Theory,” Journal of Sound and Vibration, Vol. 187, No. 5, 1995, pp. 737-756. doi:10.1006/jsvi.1995.0560

[12] A. A. Khdeir and J. N. Reddy, “An Exact Solution for the Bending of Thin and Thick Cross-Ply Laminated Beams,” Computers and Structures, Vol. 37, 1997, pp. 195-203. doi:10.1016/S0263-8223(97)80012-8

[13] V. Yildirm, E. Sancaktar and E. Kiral, “Comparison of the In-Plan Natural Frequencies of Symmetric Cross-Ply Laminated Beams Based On The Bernoulli-Eurler and Timoshenko Beam Theories,” Journal of Applied Mechanics, Vol. 66, No. 2, 1999, pp. 410-417. doi:10.1115/1.2791064

[14] A. Chakraborty, D. R. Mahapatra and S. Gopalakrishnan, “Finite Element Analysis of Free vibration and Wave Propagation in Asymmetric Composite Beams With Struc tural Discontinuities,” Composite Structures, Vol. 55, No. 1, 2002, pp. 23-36. doi:10.1016/S0263-8223(01)00130-1

[15] M. Eisenberger, “An Exact High Order Beam Element,” Computer and Structures, Vol. 81, No. 3, 2003, pp. 147 152. doi:10.1016/S0045-7949(02)00438-8

[16] J. Lee and W. W. Schultz, “Eigenvalue Analysis of Timoshenko Beams and Axi-symmetric Mindlin Plates by the Pseudospectral Method,” Journal of Sound and Vibration, Vol. 269, No. 3-5, 2004, pp. 609-621. doi:10.1016/S0022-460X(03)00047-6

[17] P. Subramanian, “Dynamic Analysis of Laminated Composite Beams Using Higher Order Theories and Finite Elements,” Composite and Structures, Vol. 73, No. 3, 2006, pp. 342-353. doi:10.1016/j.compstruct.2005.02.002

[18] M. Simsek and T. Kocaturk, “Free Vibration Analyses of Beams by Using a Third-Order Shear Deformation Theory,” Sadana, Vol. 32, Part 3, 2007, pp. 167-179. doi:10.1007/s12046-007-0015-9

[19] L. Jun, H. Hongxinga and S. Rongyinga, “Dynamic Finite Element Method for Generally Laminated Composite Beams,” International Journal of Mechanical Sciences, Vol. 50, No. 3, 2008, pp. 466-480. doi:10.1016/j.ijmecsci.2007.09.014

[20] U. Lee and I. Jang, “Spectral Element Model for Axially Loaded Bending-Shear-Torsion Coupled Composite Timoshenko Beams,” Composite Structures, Vol. 92, No. 12, 2010, pp. 2860-2870. doi:10.1016/j.compstruct.2010.04.012

[21] Q. H. Nguyen, E. Martinellib and M. Hjiaja, “Derivation of the Exact Stiffness Matrix for a Two-Layer Timoshenko Beam Element with Partial Interaction,” Engineering Structures, Vol. 33, No. 2, 2011, pp. 298-307. doi:10.1016/j.engstruct.2010.10.006

[22] X. Lina and Y. X. Zhang, “A Novel One-Dimensional Two-Node Shear-Flexible Layered Composite Beam Element,” Finite Elements in Analysis and Design, Vol. 47, No. 7, 2011, pp. 676-682. doi:10.1016/j.finel.2011.01.010

[23] G. J. Kennedya, J. S. Hansena and J. R. R. A Martinsb, “A Timoshenko Beam Theory with Pressure Corrections for Layered Orthotropic Beams,” International Journal of Solids and Structures, Vol. 48, No. 16-17, 2011, pp. 2373 2382. doi:10.1016/j.ijsolstr.2011.04.009

[24] J. N. Reddy, “An Introduction to Nonlinear Finite Ele ment Analysis,” Oxford University Press, USA, 2004.

[25] J. N. Reddy, “Mechanics of Laminated Composite Plates and Shells-Theory and Analysis,” 2nd Edition, CRC Press, USA, 2004.

[26] S. S. Rao, “The Finite Element Method in Engineering,” 2nd Edition, BPCC Wheatons Ltd., Exeter, 1989, pp. 206-207.

[27] R. D. Cook, D. S. Malkus and M. E. Plesha, “Concept and Applications of Finite Element Analysis,” 3rd Edition, John Wiley & Sons, 1974, p. 96.

[28] W. C. Hurty and M. F. Rubinstein, “Dynamics of Structures,” Prentice Hall, New Delhi, 1967.

[29] T. Kocaturk and M. Simsek, “Free Vibration Analysis of Timoshenko Beams under Various Boundary Conditions,” Sigma Journal of Engineering and Natural Science, Vol. 1, 2005, pp. 30-44.

[30] K. Chandrashekhara, K. Krishnamurthy and S. Roy, “Free Vibration of Composite Beams Using a Refined Shear Flexible Beam Element,” Computers and Structures, Vol. 14, No. 4, 1990, pp. 269-279. doi:10.1016/0263-8223(90)90010-C

[1] F. G. Yuan and R. E. Miller, “A New Finite Element for Laminated Composite Beams,” Computers and Structures, Vol. 31, No. 5, 1989, pp. 737-745. doi:10.1016/0045-7949(89)90207-1

[2] K. Chandrashekhara and K. M. Bangera, “Free Vibration of Composite Beams Using a Refined Shear Flexible Beam Element,” Computers and Structures, Vol. 43, No. 4, 1992, pp. 719-727. doi:10.1016/0045-7949(92)90514-Z

[3] Z. Friedman and J. B. Kosmatka, “An Improved Two Node Timoshenko Beam Finite Element,” Computer and Structures, Vol. 47, No. 3, 1993, pp. 473-481. doi:10.1016/0045-7949(93)90243-7

[4] T. Lidstrom, “An Analytical Energy Expression for Equilibrium Analysis of 3-D Timoshenko Beam Element,” Computers and Structures, Vol. 52, No. 1, 1994, pp. 95 101. doi:10.1016/0045-7949(94)90259-3

[5] S. R. Bhate, U. N. Nayak and A. V. Patki, “Deformation of Composite Beam Using Refined Theory,” Computers and Structures, Vol. 54, No. 3, 1995, pp. 541-546. doi:10.1016/0045-7949(94)00354-6

[6] S. M. Nabi and N. Ganesan, “Generalized Element for the Free Vibration Analysis of Composite Beams,” Computers and Structures, Vol. 51, No.5, 1994, pp. 607-610. doi:10.1016/0045-7949(94)90068-X

[7] E. A. Armanios and A. M. Badir, “Free Vibration Analysis of the Anisotropic Thin-Walled Closed-Section Beams,” Journal of the American Institute of Aeronautics and Astronautics, Vol. 33, No. 10, 1995, pp. 1905-1910. doi:10.2514/3.12744

[8] V. Giavotto, M. Borri, P. Mantegazza., G. Ghiringhelli, V. Carmash, G. Maffioli and F. Mussi, “Anisotropic Beam Theory And Applications,” Computers and Structures, Vol. 16, No. 4, 1983, pp. 403-413. doi:10.1016/0045-7949(83)90179-7

[9] D. Hagodes, A. Atilgan, M. Fulton and L. Rehfield, “Free Vibration Analysis of Composite Beams,” Journal of the American Helicopter Society, Vol. 36, No. 3, 1991, pp. 36-47. doi:10.4050/JAHS.36.36

[10] R. Chandra and I. Chopra, “Experimental and Theoretical Investigation of the Vibration Characteristics of Rotating Composite Box Beams,” Journal of Aircraft, Vol. 29, No. 4, 1992, pp. 657-664. doi:10.2514/3.46216

[11] S. R. Rao and N. Ganesan, “Dynamic Response of Tapered Composites Beams Using Higher Order Shear Deformation Theory,” Journal of Sound and Vibration, Vol. 187, No. 5, 1995, pp. 737-756. doi:10.1006/jsvi.1995.0560

[12] A. A. Khdeir and J. N. Reddy, “An Exact Solution for the Bending of Thin and Thick Cross-Ply Laminated Beams,” Computers and Structures, Vol. 37, 1997, pp. 195-203. doi:10.1016/S0263-8223(97)80012-8

[13] V. Yildirm, E. Sancaktar and E. Kiral, “Comparison of the In-Plan Natural Frequencies of Symmetric Cross-Ply Laminated Beams Based On The Bernoulli-Eurler and Timoshenko Beam Theories,” Journal of Applied Mechanics, Vol. 66, No. 2, 1999, pp. 410-417. doi:10.1115/1.2791064

[14] A. Chakraborty, D. R. Mahapatra and S. Gopalakrishnan, “Finite Element Analysis of Free vibration and Wave Propagation in Asymmetric Composite Beams With Struc tural Discontinuities,” Composite Structures, Vol. 55, No. 1, 2002, pp. 23-36. doi:10.1016/S0263-8223(01)00130-1

[15] M. Eisenberger, “An Exact High Order Beam Element,” Computer and Structures, Vol. 81, No. 3, 2003, pp. 147 152. doi:10.1016/S0045-7949(02)00438-8

[16] J. Lee and W. W. Schultz, “Eigenvalue Analysis of Timoshenko Beams and Axi-symmetric Mindlin Plates by the Pseudospectral Method,” Journal of Sound and Vibration, Vol. 269, No. 3-5, 2004, pp. 609-621. doi:10.1016/S0022-460X(03)00047-6

[17] P. Subramanian, “Dynamic Analysis of Laminated Composite Beams Using Higher Order Theories and Finite Elements,” Composite and Structures, Vol. 73, No. 3, 2006, pp. 342-353. doi:10.1016/j.compstruct.2005.02.002

[18] M. Simsek and T. Kocaturk, “Free Vibration Analyses of Beams by Using a Third-Order Shear Deformation Theory,” Sadana, Vol. 32, Part 3, 2007, pp. 167-179. doi:10.1007/s12046-007-0015-9

[19] L. Jun, H. Hongxinga and S. Rongyinga, “Dynamic Finite Element Method for Generally Laminated Composite Beams,” International Journal of Mechanical Sciences, Vol. 50, No. 3, 2008, pp. 466-480. doi:10.1016/j.ijmecsci.2007.09.014

[20] U. Lee and I. Jang, “Spectral Element Model for Axially Loaded Bending-Shear-Torsion Coupled Composite Timoshenko Beams,” Composite Structures, Vol. 92, No. 12, 2010, pp. 2860-2870. doi:10.1016/j.compstruct.2010.04.012

[21] Q. H. Nguyen, E. Martinellib and M. Hjiaja, “Derivation of the Exact Stiffness Matrix for a Two-Layer Timoshenko Beam Element with Partial Interaction,” Engineering Structures, Vol. 33, No. 2, 2011, pp. 298-307. doi:10.1016/j.engstruct.2010.10.006

[22] X. Lina and Y. X. Zhang, “A Novel One-Dimensional Two-Node Shear-Flexible Layered Composite Beam Element,” Finite Elements in Analysis and Design, Vol. 47, No. 7, 2011, pp. 676-682. doi:10.1016/j.finel.2011.01.010

[23] G. J. Kennedya, J. S. Hansena and J. R. R. A Martinsb, “A Timoshenko Beam Theory with Pressure Corrections for Layered Orthotropic Beams,” International Journal of Solids and Structures, Vol. 48, No. 16-17, 2011, pp. 2373 2382. doi:10.1016/j.ijsolstr.2011.04.009

[24] J. N. Reddy, “An Introduction to Nonlinear Finite Ele ment Analysis,” Oxford University Press, USA, 2004.

[25] J. N. Reddy, “Mechanics of Laminated Composite Plates and Shells-Theory and Analysis,” 2nd Edition, CRC Press, USA, 2004.

[26] S. S. Rao, “The Finite Element Method in Engineering,” 2nd Edition, BPCC Wheatons Ltd., Exeter, 1989, pp. 206-207.

[27] R. D. Cook, D. S. Malkus and M. E. Plesha, “Concept and Applications of Finite Element Analysis,” 3rd Edition, John Wiley & Sons, 1974, p. 96.

[28] W. C. Hurty and M. F. Rubinstein, “Dynamics of Structures,” Prentice Hall, New Delhi, 1967.

[29] T. Kocaturk and M. Simsek, “Free Vibration Analysis of Timoshenko Beams under Various Boundary Conditions,” Sigma Journal of Engineering and Natural Science, Vol. 1, 2005, pp. 30-44.

[30] K. Chandrashekhara, K. Krishnamurthy and S. Roy, “Free Vibration of Composite Beams Using a Refined Shear Flexible Beam Element,” Computers and Structures, Vol. 14, No. 4, 1990, pp. 269-279. doi:10.1016/0263-8223(90)90010-C