THE SOLUTION OF THE GOVERNING EQUATION OF THE BEAM ON LINEAR SPRING FOUNDATION MODELED BY A DISCONTINUITY FUNCTION

Yıl 2019, Cilt: 37 Sayı: 2, 495 - 506, 01.06.2019

Duygu Dönmez Demir B. Gültekin Sınır Emine Kahraman

Öz

The structural engineering researches have attracted considerable attention by many scientist for several decades. Determining the dynamical behaviors of structural elements with some discontinuous is of great importance in many engineering applications. The mentioned structures can be modelled two different ways. In the first approximation so-called the classical approach, a fourth order differential equation are written for each part of beam separated in the distinct discontinuity locations. Therefore, we obtain a system of equation containing number of the differential equation with boundary and transient conditions. Secondly, the real problem can be reformulated by only one differential equation having discontinuity function. In this study, we introduce the method of multiple scales as the solution technique. Since we encountered by the differential equation with discontinuity function in the part of order discretization during the perturbative solution, we have used a numerical technique for the solution. The mentioned technique is applied on the beam model lying on lineer spring foundation called as Winkler type foundation.

Anahtar Kelimeler

Discontinuous structural elements, finite differences method, method of multiple scales, linear spring foundation, Winkler type foundation.

Kaynakça

• [1] Dutta, SC, Roy, R: A critical review on idealization and modeling for interaction among soil-foundation-structure system. Computers & Structures, 80(20-21), 1579-1594 (2002).
• [2] Wang, CM, Reddy, JN, Lee, KH: Shear deformable beams and plates, relationships with classical solutions. Elsevier (2000).
• [3] Hayir, A: Dynamic behavior of an elastic beam on a Winkler foundation under a moving load. International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, 4, 1-4 (2010).
• [4] Friswell, MI, Penny, JET: A simple nonlinear model of a cracked beam. Proc. 10th Int. Modal Analysis Conf., San Diego, U.S.A., 1, 516-521 (1992).
• [5] Failla, G: On the dynamics of viscoelastic discontinuous beams. Mechanics Research Communications, 60, 52-63 (2014).
• [6] Failla, G, Santini, A: A solution method for Euler–Bernoulli vibrating discontinuous beams. Mechanics Research Communications, 35, 517-529 (2008).
• [7] Failla, G, Santini, A: On Euler–Bernoulli discontinuous beam solutions via uniform-beam Green’s functions. International Journal of Solids and Structures, 44, 7666–7687 (2007).
• [8] Li, QS: Free vibration analysis of non-uniform beams with an arbitrary number of cracks and concentrated masses. Journal of Sound and Vibration, 252(3), 509-525 (2002).
• [9] Dinev, D: Analytical solution of beam on elastic foundation by singularity functions. Engineering Mechanics, 19(5), 1-12 (2012).
• [10] Basu, D, Kameswara Rao, NSV: Analytical solutions for Euler–Bernoulli beam on visco‐elastic foundation subjected to moving load. Int. J. Numer. Anal. Meth. Geomech., 37, 945-960 (2013).
• [11] M. Attar, M, Karrech, A, Regenauer-Lieb, K: Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model. Journal of Sound and Vibration, 333, 2359-2377 (2014).
• [12] Sınır, BG, Çetin, K, Usta, L: Effect of soil coefficients and Poisson’s ratio on the behavior of modified Euler-Bernoulli beam lying on Winkler foundation. International Journal of Computational and Experimental Science and Engineering, 3(2), 50-53 (2017).
• [13] Nayfeh, AH: Introduction to perturbation techniques. New York: Wiley (1981).