Transfer matrisi kullanılarak Pasternak zemini üzerine oturan kademeli Timoshenko kirişlerinin serbest titreşim analizi

Bu çalışmada, iki parametreli elastik zemin üzerine oturan kademeli Timoshenko kirişlerinin serbest titreşim analizinin transfer matrisi yöntemi (TMM) kullanılarak gerçekleştirilmesi amaçlanmıştır. Winkler yaylarına eklenmiş, sıkıştırılamaz dikey elemanlardan oluşan bir kayma tabakasına sahip olan Pasternak zemin modeli dikkate alınmıştır. Pasternak zemini üzerine oturan Timoshenko kirişlerinin hareket denklemlerinin kapalı çözümlerine dayanan transfer matrisi formülasyonları elde edilmiştir. Doğal frekanslar, sınır şartlarına göre indirgeme yapılması sonrası yapının global transfer matrisinin determinantının sıfıra eşitlenmesiyle hesaplanmıştır. Mod şekilleri, uçlardaki durum vektörleri normalize edilerek çizilmiştir. İlk olarak, her iki ucu basit mesnetli bir kiriş için, önerilen yaklaşım kullanılarak elde edilen doğal frekanslar literatürdeki verilerle doğrulanmıştır ve çok iyi bir uyum gözlemlenmiştir. Daha sonra, sayısal analiz için basit-basit (S-S), basit-ankastre (SF), ankastre-basit (F-S) ve ankastre-ankastre (F-F) sınır koşullarına sahip üç kademeli kiriş modelleri dikkate alınmıştır. Kademeli kiriş modeli için TMM kullanılarak hesaplanan doğal frekanslar, elastik zeminin kayma tabakasının etkileri ihmal edilerek SAP2000'in sonlu elemanlar yöntemi (FEM) sonuçlarıyla karşılaştırılmıştır. Winkler yaylarının rijitliğinin ve kayma tabakasının kademeli kiriş modelinin doğal frekansları üzerindeki etkileri sırasıyla S-S, S-F, F-S ve F-F sınır koşulları için ortaya çıkarılmıştır. Kademeli kiriş modelinin mod şekilleri sunulmuştur. Sonuçlar, TMM'nin Pasternak zemini üzerine oturan çok kademeli Timoshenko kirişlerinin serbest titreşim analizi için etkili bir araç olarak kullanılabileceğini göstermektedir.

Free vibration analysis of segmented Timoshenko beams on Pasternak foundation by using transfer matrix

The aim of this study is performing free vibration analysis of segmented Timoshenko beams on twoparameter elastic foundation by using the transfer matrix method (TMM). The Pasternak foundationmodel which has an incompressible shear layer of vertical elements attached to the Winkler springswas considered. The transfer matrix formulations which are based on closed-form solutions ofequations of motion of Timoshenko beams on Pasternak foundation were obtained. The naturalfrequencies were calculated by equating the determinant of global transfer matrix of structure to zeroafter the reduction according to boundary conditions. The mode shapes were plotted by normalisingthe state vectors at the ends. Firstly, the natural frequencies that obtained by using the proposedapproach were validated by data in literature for a simply supported beam where a very goodagreement was observed. Then, three-segmented beam models having various boundary conditionsnamely simple-simple (S-S), simple-fixed (S-F), fixed-simple (F-S) and fixed-fixed (F-F) wereconsidered for numerical analysis. For the segmented beam models, the natural frequencies thatcalculated by using the TMM were compared to the results of finite element method (FEM) fromSAP2000 by ignoring effects of shear layer of elastic foundation. The effects of shear layer as well asstiffness of Winkler springs on the natural frequencies of the segmented beam model were revealedfor S-S, S-F, F-S and F-F boundary conditions, respectively. The mode shapes of the segmented beammodel were presented. The results show that TMM can be used as an effective tool for free vibrationanalysis of multi-segmented Timoshenko beams on Pasternak foundation.

PDF

___

[1] Lee, J.K., Jeong, S., Lee, J. 2014. Natural frequencies for flexural and torsional vibrations of beams on Pasternak foundation. Soils and Foundations, Vol: 54, pp:1202-11. https://doi.org/10.1016/j.sandf.2014.11.013
[2] Eisenberger, M., Clastornik, J. 1987. Vibrations and buckling of a beam on a variable winkler elastic foundation. Journal of Sound and Vibration, Vol: 115, pp:233-41. https://doi.org/10.1016/0022- 460X(87)90469-X
[3] Zhou, D. 1993. A general solution to vibrations of beams on variable winkler elastic foundation. Computers & Structures, Vol: 47, pp:83-90. https://doi.org/10.1016/0045-7949(93)90281-H
[4] Catal, H.H. 2002. Free vibration of partially supported piles with the effects of bending moment, axial and shear force. Engineering Structures, Vol: 24, pp:1615-22. https://doi.org/10.1016/S0141- 0296(02)00113-X
[5] Catal, H.H. 2006. Free vibration of semi-rigid connected and partially embedded piles with the effects of the bending moment, axial and shear force. Engineering Structures, Vol: 28, pp:1911-8. https://doi.org/10.1016/j.engstruct.2006.03.018
[6] Yesilce, Y., Catal, H.H. 2011. Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method. Archive of Applied Mechanics, Vol: 81, pp:199-213. 10.1007/s00419-010-0405-z
[7] Mirzabeigy, A., Madoliat, R. 2016. Large amplitude free vibration of axially loaded beams resting on variable elastic foundation. Alexandria Engineering Journal, Vol: 55, pp:1107-14. https://doi.org/10.1016/j.aej.2016.03.021
[8] Calim, F.F. 2020. Vibration Analysis of Functionally Graded Timoshenko Beams on Winkler–Pasternak Elastic Foundation. Iranian Journal of Science and Technology, Transactions of Civil Engineering, Vol: 44, pp:901-20. 10.1007/s40996-019-00283-x
[9] Yokoyama, T. 1987. Vibrations and transient responses of Timoshenko beams resting on elastic foundations. Ingenieur-Archiv, Vol: 57, pp:81-90. 10.1007/BF00541382
[10] Wang, C.M., Lam, K.Y., He, X.Q. 1998. Exact Solutions for Timoshenko Beams on Elastic Foundations Using Green's Functions∗. Mechanics of Structures and Machines, Vol: 26, pp:101-13. 10.1080/08905459808945422
[11] Caliò, I., Greco, A. 2013. Free vibrations of Timoshenko beam-columns on Pasternak foundations. Journal of Vibration and Control, Vol: 19, pp:686-96. 10.1177/1077546311433609
[12] Tonzani, G.M., Elishakoff, I. Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation. Mathematics and Mechanics of Solids, Vol: 0, pp:1081286520947775. 10.1177/1081286520947775
[13] Lee, J.W., Lee, J.Y. 2016. Free vibration analysis using the transfer-matrix method on a tapered beam. Computers & Structures, Vol: 164, pp:75-82. https://doi.org/10.1016/j.compstruc.2015.11.007
[14] Feyzollahzadeh, M., Bamdad, M. 2020. A modified transfer matrix method to reduce the calculation time: A case study on beam vibration. Applied Mathematics and Computation, Vol: 378, pp:125238. https://doi.org/10.1016/j.amc.2020.125238
[15] Chen, G., Zeng, X., Liu, X., Rui, X. 2020. Transfer matrix method for the free and forced vibration analyses of multi-step Timoshenko beams coupled with rigid bodies on springs. Applied Mathematical Modelling, Vol: 87, pp:152-70. https://doi.org/10.1016/j.apm.2020.05.023
[16] Attar, M. 2012. A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. International Journal of Mechanical Sciences, Vol: 57, pp:19-33. https://doi.org/10.1016/j.ijmecsci.2012.01.010
[17] Bozyigit, B., Yesilce, Y., Wahab Magd, A. 2020. Transfer matrix formulations and single variable shear deformation theory for crack detection in beam-like structures. Structural Engineering and Mechanics, Vol: 73, pp:109-21. 10.12989/SEM.2020.73.2.109
[18] Bozyigit, B., Yesilce, Y., Wahab Magd, A. 2020. Free vibration and harmonic response of cracked frames using a single variable shear deformation theory. Structural Engineering and Mechanics, Vol: 74, pp:33-54. 10.12989/SEM.2020.74.1.033
[19] Timoshenko, S.P. 1921. LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol: 41, pp:744-6. 10.1080/14786442108636264