The analysis of beams on elastic foundation by the methods of harmonic differential quadrature and discrete singular convolution

Abstract

Static, buckling and free vibration analyses of beams on Winkler foundation are presented. For buckling and free vibration, the governing equation of beams has been solved by reducing a standard eigenvalue equation using the harmonic differential quadrature and discrete singular convolution methods. For static analysis, however, the bending equation is reduced a set of linear algebraic equation. The effects of grid numbers and some parameters on results are investigated, and the obtained results are compared with the results of produced by analytical and other numerical methods

Keywords

• Beams on elastic foundation, vibration, buckling, numerical analysis

Elastik zemine oturan kirişlerin ayrık tekil konvolüsyon ve harmonik diferansiyel quadrature yöntemleriyle analizi

Abstract

Winkler elastik zemine oturan kirişlerin statik, burkulma ve serbest titreşim hesabı sunulmuştur. Kirişe ait yönetici diferansiyel denklem burkulma ve titreşim hesabında, harmonik diferansiyel quadrature (HDQ) yöntemi ve ayrık tekil konvolüsyon yöntemleriyle standart bir özdeğer denklemine indirgenerek çözülmüştür. Statik hesap için kirişe ait eğilme denklemi bir lineer denklem takımına indirgenmiştir. Düğüm nokta sayısı ve bazı parametrelerin sonuçlar üzerine etkisi incelenmiş, elde edilen sonuçlar analitik ve diğer sayısal çözüm yöntemleriyle karşılaştırılmıştır

Keywords

• Elastik zemine oturan kiriş, titreşim, burkulma, sayısal analiz

References

1. 1] Vlasov, V.Z., and Leont’ev N.N., Beams, Plates and Shells on Elastic foundations, Translated from Russion to Enghlish by Barouch, A., Israel Program for scientific translations, Jerusalem, (1966).
2. [2] Winkler, E., Die Lehre von der Elastizitat und Festigkeit, p. 182, Prague, (1867)
3. [3] Pasternak, P.L., New method of calculation for flexible substructures on twoparameter elastic foundation, Gosudarstvennoe Izdatelstoo, Literaturi po Stroitelstvu Arkhitekture, pp. 1–56 , Moskau (in Russian), (1954).
4. [4] Zimmermann, H., Die Berechnung des Eisen bahnoberbaues, second ed. Berlin, 1930. A.B. Kerr, Elastic and viscoelastic foundation models, Journal of Applied Mechanics 31, 491–498, (1964).
5. [5] Reissner, E., A note on deflections of plates on viscoelastic foundation, Journal of Applied. Mechanics, ASME, 25 (1), 144–145 (1958).
6. [6] Ayvaz, Y., Daloğlu, A., and Doğangün, A., Applicatıon of a modified Vlasov Model to earthquake analysis of plates resting on elastic foundations, Journal of Sound and Vibration, 212 (3), 499-509, (1998).
7. [7] Daloğlu, A., Doğangün, A., and Ayvaz, Y., Dynamic analysıs of foundation plates using a consistent Vlasov Model , Journal of Sound and Vibration, 224(5), 941- 951, (1999).
8. [8] Hetenyi, M., Beams on elastic foundation, The University of Michigan Press, (1946). [9] Hetenyi, M., Beams and plates on elastic foundations and related problems, Applied Mechanics Reviews, 19, 95-102, (1966).

9. [10] Kameswara, Rao. NSV., Das, YC., Anandakrishnan M, Dynamic response of beams on generalized elastic foundation. International Journal of Solids and Structures, 11, 255-73, (1975).
10. [11] Weaver, W.JR, Timoshenko SP, Young, DH, Vibration problems in engineering, Fifth Edition, Wiley, New York, (1990).
11. [12] Ayvaz, Y., Daloğlu, A., Earthquake analysis of beams resting on elastic foundations by using a modified Vlasov Model, Journal of Sound and Vibration, 200(3), 315-325, (1997).
12. [13] Ting, B.Y., Finite beams on elastic foundation with restraints. Journal of the Structural Division ASCE, 108, 611-21, (1982).
13. [14] Lentini, M., Numerical solution of the beam equation with nonuniform foundation coefficient, Journal of Applied Mechanics ASME, 46, 901-4, (1979).
14. [15] Kadıoğlu, F., Elastik zemine oturan doğru ve daire eksenli çubukların karışık sonlu eleman yöntemi ile çözümü, Çukurova Üniversitesi 15. Yıl Sempozyumu, Adana, 4-7 Nisan, 61-73, (1994).
15. [16] Lai, Y.C., Ting, B.Y., Lee W.S., and Becker W.R., Dynamic response of beams on elastic foundation, Journal of Structural Engineering ASCE. 118,853-858, 1992.
16. [17] Wang, J., Vibration of stepped beams on elastic foundations, Journal of Sound and Vibration, 149, 315-322, (1991).
17. [18] Rosa Maria A.D., Stability and dynamics of beams on Winkler elastic foundations, Earthquake Engineering & Structural Dynamics, 18, 377-88 (1989). [19] Yankelevsky, D. Z., Eisenberger, M., Analysis of a beam-column on elastic foundations, Composite Structures. 23(3), 351-56, (1986).
18. [20] Eisenberger, M., Clastornik, J., Eisenberger, M., Vibrations and buckling of a beam on variable winkler elastic foundation, Journal of Sound and Vibration. 115(2), 233-41, (1987).
19. [21] Razaqpur, A.G., Stiffness of beam-columns on elastic foundation with exact shape functions, Composite Structures, 24(5), 813-19, (1986).
20. [22] Timoshenko S.P., Gere JM. Theory of elastic stability, 2nd ed., Tokyo: McGrawHill, (1959).
21. [23] Chajes, A., Principles of structural stability theory. New Jersey: Prentice-Hall, (1974).
22. [24] Bert, C.W., Malik, M.,(1996): Differential quadrature method in computational mechanics: A Review., “Applied Mechanics Review, 49(1), 1-28.
23. [25] Civalek, Ö., Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic Columns, Engineering Structures, An International Journal, Vol. 26(2), 171- 186, (2004).
24. [26] Civalek, Ö., Ülker, M., Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates, International Journal of Structural Engineering and Mechanics, Vol. 17(1), 1-14, (2004).
25. [27] Civalek, Ö., Ülker, M., HDQ-FD Integrated methodology for nonlinear static and dynamic response of doubly curved shallow shells, International Journal of Structural Engineering and Mechanics, 19(5), 535-550, (2005).
26. [28] Civalek, Ö., Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of HDQ-FD methods, International Journal of Pressure Vessels and Piping, 82(6), 470-479, (2005).
27. [29] Civalek, Ö., Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation, Journal of Sound and Vibration, 294, 966-980, (2006).
28. [30] Civalek, Ö., Elastik zemine oturan yapıların hesap yöntemlerine genel bir bakış, Türkiye İnşaat Mühendisleri Odası-TMH, Mühendislik Haberleri, 432, 45- 54, (2004).
29. [31] Civalek, Ö., Elastik zemine oturan kirişlerin Nöro-Fuzzy tekniği ile analizi, 7. Ulusal zemin mekaniği ve temel mühendisliği konferansı, 22-23 Ekim, Yıldız Üniversitesi., İstanbul, (1998).
30. [32] Civalek, Ö., Free vibration analysis of composite conical shells using the discrete singular convolution algorithm, Steel and Composite Structures, 6(4), 353-366, (2006).
31. [33] Civalek, Ö., Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods, Applied Mathematical Modeling, 31, 606-624, (2007).
32. [34] Civalek, Ö., Frequency analysis of isotropic conical shells by discrete singular convolution (DSC), International Journal of Structural Engineering and Mechanics, 25(1), 127-131, (2007).
33. [35] Civalek, Ö., Nonlinear dynamic response of MDOF systems by the method of harmonic differential quadrature (HDQ), International Journal of Structural Engineering and Mechanics, 25 (2), 201-217, (2007).
34. [36] Civalek, Ö.,Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method, International Journal of Mechanical Sciences, 49-752-765, (2007).
35. [37] Civalek, Ö., Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: discrete singular convolution (DSC) approach, Journal of Computational and Applied Mathematics, 205, 251–271, (2007).
36. [38] Civalek, Ö., Vibration analysis of conical panels using the method of discrete singular convolution, Communications in Numerical Methods in Engineering, 24, 169-181, (2008)
37. [39] Civalek, Ö., A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution, Thin-Walled Structures, 45, 692-698, (2007).
38. [40]Civalek, Ö., Free vibration and buckling analyses of composite plates with straightsided quadrilateral domain based on DSC approach, Finite Elements in Analysis and Design, 43,1013-1022, (2007).