Abstract
The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Wang C.M., Reddy J.N., Lee K.H.: Shear Deformable Beams and Plates: Relationships with Classical Solutions. Elsevier Science Ltd., The Netherlands (2000)
Timoshenko S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)
Cowper G.R.: The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33(2), 335–340 (1966)
Gruttmann F., Wagner W.: Shear coefficient factors in Timoshenko’s beam theory for arbitrary shaped cross-section. Comput. Mech. 27, 199–207 (2001)
Murthy A.V.: Vibration of short beams. AIAA 8, 34–38 (1970)
Han S.M., Benaroya H., Wei T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225, 935–988 (1999)
Bickford W.B.: A consistent higher order beam theory. Dev. Theor. Appl. Mech. 11, 137–150 (1982)
Levinson M.: A new rectangular beam theory. J. Sound Vib. 74, 81–87 (1981)
Heyliger P.R., Reddy J.N.: A higher-order beam finite element for bending and vibration problems. J. Sound Vib. 126(2), 309–326 (1988)
Reddy J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. John Wiley, New York (2002)
Reddy J.N.: Theory and Analysis of Elastic Plates and Shells, 2nd edn. Taylor & Francis, Philadelphia (2007)
Hetenyi M.: Beams on Elastic Foundations, 7th edn. The University of Michigan Press, Michigan (1955)
Doyle P.F., Pavlovic M.N.: Vibration of beams on partial elastic foundations. J. Earthquake Eng. Struct. Dyn. 10, 663–674 (1982)
West H.H., Mafi M.: Eigenvalues for beam columns on elastic supports. J. Struct. Eng. 110(6), 1305–1320 (1984)
Yokoyama T.: Vibrations of Timoshenko beam-columns on two parameter elastic foundations. Earthquake Eng. Struct. Dyn. 20, 355–370 (1991)
Esmailzadeh E., Ohadi A.R.: Vibration and stability analysis of non-uniform Timoshenko beams under axial and distributed tangential loads. J. Sound Vib. 236, 443–456 (2000)
Yesilce Y., Catal H.H.: Free vibration of piles embedded in soil having different modulus of subgrade reaction. Appl. Math. Model. 32, 889–900 (2008)
Zhou J.K.: Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press, Wuhan China (1986)
Chen C.K., Ho S.H.: Application of differential transformation to eigenvalue problem. J. Appl. Math. Comput. 79, 173–188 (1996)
Chen C.K., Ho S.H.: Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform. Int. J. Mech. Sci. 41, 1339–1356 (1999)
Jang M.J., Chen C.L.: Analysis of the response of a strongly non-linear damped system using a differential transformation technique. Appl. Math. Comput. 88, 137–151 (1997)
Chen C.L., Liu Y.C.: Solution of two-point boundary-value problems using the differential transformation method. J. Optim. Theory Appl. 99, 23–25 (1998)
Jang M.J., Chen C.L., Liu Y.C.: On solving the initial-value problems using differential transformation method. Appl. Math. Comput. 115, 145–160 (2000)
Hassan I.H.A.H.: On solving some eigenvalue problems by using differential transformation. Appl. Math. Comput. 127, 1–2 (2002)
Hassan I.H.A.H.: Different applications for the differential transformation in the differential equations. Appl. Math. Comput. 129, 183–201 (2002)
Ayaz F.: Application of differential transforms method to differential-algebraic equations. Appl. Math. Comput. 152, 648–657 (2004)
Kurnaz A., Oturanç G., Kiris M.E.: n-Dimensional differential transformation method for solving PDEs. Int. J. Comput. Math. 82(3), 369–380 (2005)
Malik M., Hao H.D.: Vibration analysis of continuous systems by differential transformation. Appl. Math. Comput. 96, 17–26 (1998)
Bert C.W., Zeng H.: Analysis of axial vibration of compound bars by differential transformation method. J. Sound Vib. 275, 641–647 (2004)
Özdemir Ö, Kaya M.O.: Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method. J. Sound Vib. 289, 413–420 (2006)
Ozgumus O.O., Kaya M.O.: Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method. Meccanica 41, 661–670 (2006)
Çatal S.: Analysis of free vibration of beam on elastic soil using differential transform method. Struct. Eng. Mech. 24(1), 51–52 (2006)
Çatal S.: Solution of free vibration equations of beam on elastic soil by using differential transform method. Appl. Math. Model. 32, 1744–1757 (2008)
Çatal S., Çatal H.H.: Buckling analysis of partially embedded pile in elastic soil using differential transform method. Struct. Eng. Mech. 24(2), 247–268 (2006)
Ho S.H., Chen C.K.: Free transverse vibration of an axially loaded non-uniform sinning twisted Timoshenko beam using differential transform. Int. J. Mech. Sci. 48, 1323–1331 (2006)
Bildik N., Konuralp A., Bek F.O., Küçükarslan S.: Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method. Appl. Math. Comput. 172, 551–567 (2006)
Ozgumus O.O., Kaya M.O.: Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending-torsion coupling. Int. J. Eng. Sci. 45, 562–586 (2007)
Kaya M.O., Ozgumus O.O.: Flexural–torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM. J. Sound Vib. 306, 495–506 (2007)
Ertürk V.S., Momani S.: Comparing numerical methods for solving fourth-order boundary value problems. Appl. Math. Comput. 188, 1963–1968 (2007)
Ertürk V.S.: Application of differential transformation method to linear sixth-order boundary value problems. Appl. Math. Sci. 1, 51–58 (2007)
Rajasekaran S.: Buckling of fully and partially embedded non-prismatic columns using differential quadrature and differential transformation methods. Struct. Eng. Mech. 28(2), 221–238 (2008)
Monforton G.R., Wu T.S.: Matrix analysis of semi-rigid connected frames. J. Struct. Div. ASCE 89(6), 14–22 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yesilce, Y., Catal, H.H. Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method. Arch Appl Mech 81, 199–213 (2011). https://doi.org/10.1007/s00419-010-0405-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-010-0405-z