Numerical free vibration analysis of higher-order shear deformable beams resting on two-parameter elastic foundation
Document Type : Research Paper
Authors
School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran
Abstract
Free vibration analysis of higher-order shear deformation beam resting on one- and two-parameter elastic
foundation is studied using differential transform method (DTM) as a part of a calculation procedure. First,
the governing differential equations of beam are derived in a general form considering the shear-free
boundary conditions (zero shear stress conditions at the top and bottom of a beam). Using DTM the derived
equations governing beams, followed by higher-order shear deformation model, Timoshenko model and
Bernoulli-Euler model are transformed to algebraic forms and a set of recurrence formulae is then derived.
Upon imposing the boundary conditions of the beam at hand, a set of algebraic equations are obtained and
expressed in matrix form. Finally, the transverse natural frequencies of the beam are calculated through an
iterative procedure. Several numerical examples have been carried out to demonstrate the competency of
the present method and the results obtained by DTM are in good agreement with those in the literature.
Afterward, the free vibration of beams followed up by different models (i.e. Bernoulli-Euler, Timoshenko
and different higher-order models) are taken into consideration.
Keywords
- differential transform method (DTM)
- elastic foundation
- Free Vibration
- higher-order beam theory (HOBT)
Main Subjects
[1].Soldatos, K.P., Selvadurai, A.P.S. (1985). Flexure of beams resting on hyperbolic elastic foundations, Solids Struct. 21(4): 373-388.
[2].Eisenberger, M., Reich, Y. (1983). Static vibration and stability analysis of non-uniform beams, Comput. Struct. 31(4), 563-571.
[3].Zhaohua, F., Cook, R.D., (?). Beam elements on two-parameter elastic foundations, J. Eng. Mech. 109(6), 1390–1402.
[4].Lee, S.Y., KE, H.Y. (1990). Free vibrations of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints, Comput. Struct. 34( 3): 421-429.
[5].Attarnejad, R., Shahba, A., Eslaminia, M. (2011). Dynamic basic displacement functions for free vibration analysis of tapered beams, J. Vib. Control 17(14): 2222-2238.
[6].Timoshenko, S.P. (1922). On the transverse vibration of bars of uniform cross-section, Philos Mag 43(253): 125-131.
[7].Cowper, G.R. (1996). The Shear Coefficient in Timoshenko’s Beam Theory, J. Appl. Mech. 33(2): 335-340.
[8].Heiliger, P.R., Reddy, J.N. (1988). A Higher Order Beam Finite Element for Bending and Vibrations Problems, J. Sound Vib. 126(2): 309-326.
[9].Morfidis, K. (2010). Vibration of Timoshenko beams on three-parameter elastic foundation. Comput. Struct. 88(5-6): 294–308.
[10]. Yihua, M., Li, O., Hongzhi, Z. (2009). Vibration Analysis of Timoshenko Beams on a Nonlinear Elastic Foundatin, Tsinghua Sci. Technol. 14(3): 322–326.
[11]. Lee, H.P. (1998). Dynamic Response of a Timoshenko Beam on a Winkler Foundation Subjected to a Moving Mass, Appl. Acoustics 55(3): 203-215.
[12]. Yokoyam, T. (1988). Parametric instability of Timoshenko beams Resting on an elastic foundation, Comput. Struct. 28(2): 207–216.
[13]. Ruta, P. (2006). The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem, J. Sound Vib. 296(1-2): 243–263.
[14]. Attarnejad, R., Shahba, A., Jandaghi Semnani, S. (2010). Application of differential transform in free vibration analysis of Timoshenko beams resting on two-parameter elastic foundation, AJSE 35(2B): 121-128.
[15]. Esmailzadeh, E., Ghorashi, M. (1997). Vibration analysis of a Timoshenko beam subjected to a travelling mass, J. Sound Vib. 199(4): 615-628.
[16]. Lin, S.C., Hsiao, K.M. (2001). Vibration analysis of a rotating Timoshenko beam, J. Sound Vib. 240(2): 303–322.
[17]. Yokoyama, T. (1996). Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations. Comput. Struct. 61(6): 995–1007.
[18]. Levinson, M. (1981). A new rectangular beam theory, J. Sound Vib. 74(1): 81-87.
[19]. Bickford, W.B. (1982). A consistent higher order beam theory, Dev. Theor. Appl. Mech., 11: 137-150.
[20]. Wang, X.D., Shi, G. (2012). Boundary Layer Solutions Induced by Displacement Boundary Conditions of Shear Deformable Beams and Accuracy Study of Several Higher-Order Beam Theories, J. Eng. Mech. 138(11): 1388-1399.
[21]. Reddy, J.N. (1984). A simple higher order theory for laminated composite plates, ASME J. Appl. Mech. 51(4): 745-752.
[22]. Wang, M.Z., Wang, W. (2003). A refined theory of beams, J. Eng. Mech. Suppl. 324-327.
[23]. Gao, Y., Wang, M. (2006). The refined theory of rectangular deep beams based on general solutions of elasticity, Sci in China Ser G 36(3): 286-297.
[24]. Bhimaraddi, A., Chandrashekhara, K. (1993). Observations on higher-order beam theory, J. Aerospace Eng. 6(4): 408–413.
[25]. Chakrabarti, A., Sheikh, A.H., Griffith, M., Oehlers, D.J. (2013). Dynamic Response of Composite Beams with Partial Shear Interaction Using a Higher-Order Beam Theory, J. Struct. Eng. 139(1): 47–56.
[26]. Lam, K.Y., Wang, C.M., He, X.Q. (2000). Canonical exact solutions for Levyplates on a two-parameter foundation using Green’s functions, J. Eng. Struct. 22(4): 364–378.
[27]. Eisenberger, M. (2003). Dynamic stiffness vibration analysis using a high-order beam model, Int. J. Numer. Meth. Eng. 57(11): 1603–1614.
[28]. Heyliger, P.R., Reddy, J.N. (1988). A higher order beam finite element for bending And vibration problems, J. Sound Vib. 126(2): 309–326.
[29]. Matsunaga, H. (1999). Vibration and buckling of deep beam-columns on two-parameter elastic foundations, J. Sound Vib. 228(2): 359–376.
[30]. Winkler, E. (1867). Die Lehre Von Der Elastizitat Und Festigkeit, Prague : Dominicus.
[31]. Pasternak, P.L. (1954). On a New Method of Analysis of an Elastic Foundation by Means of Two-Constants, USSR: Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture [in Russian], Moscow.
[32]. Filonenko–Borodich (1940). Some Approximate Theories of Elastic Foundation, Uchenyie Zapiski Moskovskogo
[2].Eisenberger, M., Reich, Y. (1983). Static vibration and stability analysis of non-uniform beams, Comput. Struct. 31(4), 563-571.
[3].Zhaohua, F., Cook, R.D., (?). Beam elements on two-parameter elastic foundations, J. Eng. Mech. 109(6), 1390–1402.
[4].Lee, S.Y., KE, H.Y. (1990). Free vibrations of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints, Comput. Struct. 34( 3): 421-429.
[5].Attarnejad, R., Shahba, A., Eslaminia, M. (2011). Dynamic basic displacement functions for free vibration analysis of tapered beams, J. Vib. Control 17(14): 2222-2238.
[6].Timoshenko, S.P. (1922). On the transverse vibration of bars of uniform cross-section, Philos Mag 43(253): 125-131.
[7].Cowper, G.R. (1996). The Shear Coefficient in Timoshenko’s Beam Theory, J. Appl. Mech. 33(2): 335-340.
[8].Heiliger, P.R., Reddy, J.N. (1988). A Higher Order Beam Finite Element for Bending and Vibrations Problems, J. Sound Vib. 126(2): 309-326.
[9].Morfidis, K. (2010). Vibration of Timoshenko beams on three-parameter elastic foundation. Comput. Struct. 88(5-6): 294–308.
[10]. Yihua, M., Li, O., Hongzhi, Z. (2009). Vibration Analysis of Timoshenko Beams on a Nonlinear Elastic Foundatin, Tsinghua Sci. Technol. 14(3): 322–326.
[11]. Lee, H.P. (1998). Dynamic Response of a Timoshenko Beam on a Winkler Foundation Subjected to a Moving Mass, Appl. Acoustics 55(3): 203-215.
[12]. Yokoyam, T. (1988). Parametric instability of Timoshenko beams Resting on an elastic foundation, Comput. Struct. 28(2): 207–216.
[13]. Ruta, P. (2006). The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem, J. Sound Vib. 296(1-2): 243–263.
[14]. Attarnejad, R., Shahba, A., Jandaghi Semnani, S. (2010). Application of differential transform in free vibration analysis of Timoshenko beams resting on two-parameter elastic foundation, AJSE 35(2B): 121-128.
[15]. Esmailzadeh, E., Ghorashi, M. (1997). Vibration analysis of a Timoshenko beam subjected to a travelling mass, J. Sound Vib. 199(4): 615-628.
[16]. Lin, S.C., Hsiao, K.M. (2001). Vibration analysis of a rotating Timoshenko beam, J. Sound Vib. 240(2): 303–322.
[17]. Yokoyama, T. (1996). Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations. Comput. Struct. 61(6): 995–1007.
[18]. Levinson, M. (1981). A new rectangular beam theory, J. Sound Vib. 74(1): 81-87.
[19]. Bickford, W.B. (1982). A consistent higher order beam theory, Dev. Theor. Appl. Mech., 11: 137-150.
[20]. Wang, X.D., Shi, G. (2012). Boundary Layer Solutions Induced by Displacement Boundary Conditions of Shear Deformable Beams and Accuracy Study of Several Higher-Order Beam Theories, J. Eng. Mech. 138(11): 1388-1399.
[21]. Reddy, J.N. (1984). A simple higher order theory for laminated composite plates, ASME J. Appl. Mech. 51(4): 745-752.
[22]. Wang, M.Z., Wang, W. (2003). A refined theory of beams, J. Eng. Mech. Suppl. 324-327.
[23]. Gao, Y., Wang, M. (2006). The refined theory of rectangular deep beams based on general solutions of elasticity, Sci in China Ser G 36(3): 286-297.
[24]. Bhimaraddi, A., Chandrashekhara, K. (1993). Observations on higher-order beam theory, J. Aerospace Eng. 6(4): 408–413.
[25]. Chakrabarti, A., Sheikh, A.H., Griffith, M., Oehlers, D.J. (2013). Dynamic Response of Composite Beams with Partial Shear Interaction Using a Higher-Order Beam Theory, J. Struct. Eng. 139(1): 47–56.
[26]. Lam, K.Y., Wang, C.M., He, X.Q. (2000). Canonical exact solutions for Levyplates on a two-parameter foundation using Green’s functions, J. Eng. Struct. 22(4): 364–378.
[27]. Eisenberger, M. (2003). Dynamic stiffness vibration analysis using a high-order beam model, Int. J. Numer. Meth. Eng. 57(11): 1603–1614.
[28]. Heyliger, P.R., Reddy, J.N. (1988). A higher order beam finite element for bending And vibration problems, J. Sound Vib. 126(2): 309–326.
[29]. Matsunaga, H. (1999). Vibration and buckling of deep beam-columns on two-parameter elastic foundations, J. Sound Vib. 228(2): 359–376.
[30]. Winkler, E. (1867). Die Lehre Von Der Elastizitat Und Festigkeit, Prague : Dominicus.
[31]. Pasternak, P.L. (1954). On a New Method of Analysis of an Elastic Foundation by Means of Two-Constants, USSR: Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture [in Russian], Moscow.
[32]. Filonenko–Borodich (1940). Some Approximate Theories of Elastic Foundation, Uchenyie Zapiski Moskovskogo
Gosudarstvennogo Universiteta Mekhanica [in Russian], 46: 3–18,
[33]. Hetenyi, M. (1946). Beams on Elastic Foundations, Ann. Arbor. Mich. USA: The University of Michigan Press.
[34]. Hetenyi, M. (1950). A general solution for the bending of beams on an elastic foundation of arbitrary, J Appl. Phys. 21(1): 55-58.
[35]. Vlasov, V.Z., Leontev, U.N. (?). Beams, Plates and Shells on Elastic Foundations. NASA XT F-357 TT 65-50135.
[36]. Catal, S. (2008). Solution of Free Vibration Equations of Beams on Elastic Soil by Using Differential Transform Method, Appl. Math. Model. 32(9): 1744–1757.
[37]. Ho, S.H., Chen, C.K. (1998). Analysis of General Elastically End Restrained Non-Uniform Beams Using Differential Transform, Appl. Math. Model. 22(4-5): 219–234.
[38]. Sayyad, A.S. (2011). Comparison of various refined beam theories for the bending and free vibration analysis of thick beams. Appl. Comput. Mech. 5(2): 217-230.
[39]. Zhao, J.K. (1988). Differential transformation and its application for electrical circuits, Huazhong.
[40]. Chen, C.K., Ho, S.H. (1996). Application of differential transformation eigenvalue problems, Appl. Math. Comput. 79(2-3): 171-179.
[41]. Kaya, M.O. (2006). Free vibration analysis of a rotating timoshenko beam by differential transform method, Aircr Eng Aerosp Tec 78(3): 194-203.
[42]. Ozgumus, O.O., Kaya, M.O. (2006). Flapwise bending vibration analysis of a rotating tapered cantilever bernoulli–euler beam by differential transform method, J. Sound Vib. 289(1-2): 413-420.
[43]. Yalcin, S., Arikoglu, A., Ozkol, I. (2009). Free vibration analysis of circular plates by differential transformation method, Appl. Mathematics and Computation 212(2): 377-386.
[44]. Attarnejad, R., Shahba, A. (2008). Application of differential transform method in free vibration analysis of rotating non-prismatic beams, World Appl. Sci. J. 5(4): 441-448.
[45]. Shahba, A., Rajasekaran, S. (2011). Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials, Appl. Math. Model. 36(7): 3094–3111.
[46]. Rajasekaran, S. (2013). Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach, Meccanica 48(5): 1053–1070.
[47]. Semnani, S.J., Attarnejad, R., Firouzjaei, R.K. (2013). Free Vibration Analysis of Variable Thickness Thin Plates by Two – dimensional Differential Transform Method, Acta Mechanica 224(8): 1643-1658.
[48]. Reddy, J.N. (2002). Energy principles and variational methods in applied mechanic. Wiley.
[49]. De Rosa, M.A. (1995). Free vibration of Timoshenko beams on two-parameter elastic foundation. Comput. Struct. 57(1): 151–156. University Press, Wuhan, China.
[50]. Balkaya, M., Kaya, M.O., Sa˘glamer, A. (2009). Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Arch. Appl. Mech. 79(2): 135–146.
[51]. Touratier, M. (1991). An efficient standard plate theory, Int. J. Eng. Sci. 29(8): 901-916.
[52]. Soldatos, K.P. (1992). A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica 94(3-4): 195-220.
[53]. Aydogdu M. (2009). A new shear deformation theory for laminated composite plates. Compos. Struct. 89(1): 94-101.
[54]. Arikoglu, A., Ozkol, I. (2007). Solution of fractional differential equations by using differential transform method, Chaos Solit. Fract. 34(5): 1473–1481.
[33]. Hetenyi, M. (1946). Beams on Elastic Foundations, Ann. Arbor. Mich. USA: The University of Michigan Press.
[34]. Hetenyi, M. (1950). A general solution for the bending of beams on an elastic foundation of arbitrary, J Appl. Phys. 21(1): 55-58.
[35]. Vlasov, V.Z., Leontev, U.N. (?). Beams, Plates and Shells on Elastic Foundations. NASA XT F-357 TT 65-50135.
[36]. Catal, S. (2008). Solution of Free Vibration Equations of Beams on Elastic Soil by Using Differential Transform Method, Appl. Math. Model. 32(9): 1744–1757.
[37]. Ho, S.H., Chen, C.K. (1998). Analysis of General Elastically End Restrained Non-Uniform Beams Using Differential Transform, Appl. Math. Model. 22(4-5): 219–234.
[38]. Sayyad, A.S. (2011). Comparison of various refined beam theories for the bending and free vibration analysis of thick beams. Appl. Comput. Mech. 5(2): 217-230.
[39]. Zhao, J.K. (1988). Differential transformation and its application for electrical circuits, Huazhong.
[40]. Chen, C.K., Ho, S.H. (1996). Application of differential transformation eigenvalue problems, Appl. Math. Comput. 79(2-3): 171-179.
[41]. Kaya, M.O. (2006). Free vibration analysis of a rotating timoshenko beam by differential transform method, Aircr Eng Aerosp Tec 78(3): 194-203.
[42]. Ozgumus, O.O., Kaya, M.O. (2006). Flapwise bending vibration analysis of a rotating tapered cantilever bernoulli–euler beam by differential transform method, J. Sound Vib. 289(1-2): 413-420.
[43]. Yalcin, S., Arikoglu, A., Ozkol, I. (2009). Free vibration analysis of circular plates by differential transformation method, Appl. Mathematics and Computation 212(2): 377-386.
[44]. Attarnejad, R., Shahba, A. (2008). Application of differential transform method in free vibration analysis of rotating non-prismatic beams, World Appl. Sci. J. 5(4): 441-448.
[45]. Shahba, A., Rajasekaran, S. (2011). Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials, Appl. Math. Model. 36(7): 3094–3111.
[46]. Rajasekaran, S. (2013). Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach, Meccanica 48(5): 1053–1070.
[47]. Semnani, S.J., Attarnejad, R., Firouzjaei, R.K. (2013). Free Vibration Analysis of Variable Thickness Thin Plates by Two – dimensional Differential Transform Method, Acta Mechanica 224(8): 1643-1658.
[48]. Reddy, J.N. (2002). Energy principles and variational methods in applied mechanic. Wiley.
[49]. De Rosa, M.A. (1995). Free vibration of Timoshenko beams on two-parameter elastic foundation. Comput. Struct. 57(1): 151–156. University Press, Wuhan, China.
[50]. Balkaya, M., Kaya, M.O., Sa˘glamer, A. (2009). Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Arch. Appl. Mech. 79(2): 135–146.
[51]. Touratier, M. (1991). An efficient standard plate theory, Int. J. Eng. Sci. 29(8): 901-916.
[52]. Soldatos, K.P. (1992). A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica 94(3-4): 195-220.
[53]. Aydogdu M. (2009). A new shear deformation theory for laminated composite plates. Compos. Struct. 89(1): 94-101.
[54]. Arikoglu, A., Ozkol, I. (2007). Solution of fractional differential equations by using differential transform method, Chaos Solit. Fract. 34(5): 1473–1481.
)
Volume 46, Issue 2 - Serial Number 2
July 2015Pages 117-131
- Receive Date: 26 November 2014
- Revise Date: 02 May 2015
- Accept Date: 02 May 2015