Abstract
In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature.
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References
Friedman Z., Kosmatka J.B.: Exact stiffness matrix of a nonuniform beam—I. Extension, torsion, and bending of a Bernoulli–Euler beam. Comput. Struct. 42, 671–682 (1992)
Kuo Y.H., Lee S.Y.: Deflection of nonuniform beams resting on a nonlinear elastic foundation. Comput. Struct. 51, 513–519 (1994)
Hosur V., Bhavikatti S.S.: Influence lines for bending moments in beams on elastic foundations. Comput. Struct. 58, 1225–1231 (1996)
Chen C.N.: Solution of beam on elastic foundation by DQEM. J. Eng. Mech. ASCE 124, 1381–1384 (1998)
Eisenberger M., Yankelevsky D.Z.: Exact stiffness matrix for beams on elastic foundation. Comput. Struct. 21, 1355–1359 (1985)
Sen Y.L., Huei Y.K., Yee H.K.: Exact static deflection of a non-uniform Bernoulli–Euler beam with general elastic end restraints. Comput. Struct. 36, 91–97 (1990)
Razaqpur A.G., Shah K.R.: Exact analysis of beams on two-parameter elastic foundations. Int. J. Solids Struct. 27, 435–454 (1991)
Morfidis K., Avramidis I.E.: Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets. Comput. Struct. 80, 1919–1934 (2002)
Kim N. II, Jeon S.-S., Kim M.-Y.: An improved numerical method evaluating exact static element stiffness matrices of thin-walled beam-columns on elastic foundations. Comput. Struct. 83, 2003–2022 (2005)
Katsikadelis J.T., Tsiatas G.C.: Optimum design of structures subjected to follower forces. Int. J. Mech. Sci. 49, 1204–1212 (2007)
Tsiatas G.C.: Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta. Mech. 209, 141–152 (2010)
Katsikadelis J.T.: The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies. Theor. Appl. Mech. 27, 13–38 (2002)
Huang Y., Li X-F.: A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J. Sound Vib. 329, 2291–2303 (2010)
Weaver W. Jr., Timoshenko S.P, Young D.H.: Vibration Problems in Engineering. Wiley, New York (1990)
Cortinez V.H., Laura P.A.A.: An extension of Timoshenko’s method and its application to buckling and vibration problems. J. Sound Vib. 169, 141–144 (1994)
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Tsiatas, G.C. A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation. Arch Appl Mech 84, 615–623 (2014). https://doi.org/10.1007/s00419-014-0820-7
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DOI: https://doi.org/10.1007/s00419-014-0820-7