Investigation of Eccentrically-Supported Simple Beam under Point Load

Öz

According to the classical beam theories, beams are considered as one dimensional element. These theories assume that supports are placed at the mid-plane of the beam. However, in practice, the beams often are supported at the point different from their centers. In this study, an eccentrically simplysupported beam under transverse point load at the center of the beam was solved by the application of the MacLaurin series. This study presents a theoretical approach to the analysis of eccentrically supported beams. The effects of eccentric supports on the flexural rigidity of the beam have been investigated. Analytic equations derived were used to investigate the effect of varying support positions through the thickness on bending analysis of beams under transverse loading. The findings revealed that the flexural rigidity of beams is significantly influenced by eccentric pin-pin support. The accuracy of the equations was verified by comparing the results obtained with the Finite Element solutions.

Anahtar Kelimeler

• Eccentric support
• Simple beam
• Beam theory
• modified beam theory
• the MacLaurin series

TEKİL YÜK ETKİSİ ALTINDA EKSANTRİK MESNETLİ KİRİŞLERİN İNCELENMESİ

Öz

Klasik kiriş teorilerinde kirişler bir boyutlu kabul edilir. Bu teoriye göre mesnetler tarafsız eksendedir. Ancak pratik uygulamalarda kirişler tarafsız eksenlerinden farklı noktalardan mesnetlenmektedir. Literatürde üniform yük etkisinde eksantrik mesnetlenme durumuna sahip kirişler için düzenlenmiş kiriş teorisi geliştirilmiştir. Bu çalışmada tekil yük altındaki eksantrik mesnetli kirişler MacLaurin serileri kullanılarak analitik olarak çözülmüştür. Eksantrik mesnetlerin kirişin eğilme rijitliği üzerindeki etkileri araştırılmıştır. Elde edilen analitik denklemler, mesnetlerin kiriş derinliğinde farklı konumlarında (eksantrisite) kirişlerin eğilme analizi üzerindeki etkisini araştırmak için kullanılmıştır. Bulgular, kirişlerin eğilme rijitliğinin eksantrik mesnet durumundan önemli ölçüde etkilendiğini göstermektedir. Elde edilen sonuçlar Sonlu Eleman çözümleri ile karşılaştırılmıştır.

Anahtar Kelimeler

• Eksantrik mesnet
• Basit kiriş
• Kiriş teorisi
• Düzenlenmiş kiriş teorisi
• MacLaurin serileri

Kaynakça

1. 1. Bernoulli, D. (1751) ‘Commentarii Academiae Scientiarum’, In: Petropoli. Chap. De vibrationibus et sono laminarum elasticarum.
2. 2. Bickford, W. B. (1982) ‘A Consistent Higher Order Beam Theory’, in And, T. C. and Karr, G. (eds) Developments in theoretical and applied mechanics: Proceedings of the eleventh southeastern conference on theoretical and applied mechanics. Huntsville, Alabama, pp. 137–150.
3. 3. Carrera, E. et al. (2015) ‘Recent developments on refined theories for beams with applications’, Mechanical Engineering Reviews, advpub. doi: 10.1299/mer.14-00298.
4. 4. Carrera, E., Valvano, S. and Kulikov, G. M. (2018) ‘Multilayered plate elements with node-dependent kinematics for electro-mechanical problems’, International Journal of Smart and Nano Materials. Taylor & Francis, 9(4), pp. 279–317. doi: 10.1080/19475411.2017.1376722.
5. 5. Dwaikat, M. and Kodur, V. (2010) ‘Effect of Location of Restraint on Fire Response of Steel Beams’, pp. 109–128. doi: 10.1007/s10694-009-0085-9.
6. 6. Eltaher, M. A., Alshorbagy, A. E. and Mahmoud, F. F. (2013) ‘Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams’, Composite Structures. Elsevier, 99, pp. 193–201. doi: 10.1016/J.COMPSTRUCT.2012.11.039.
7. 7. Euler, L. (1744) De curvis elasticis, In: Bousquet. Chap. Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accept.
8. 8. Fernando, D., Wang, C. M. and Roy Chowdhury, A. N. (2018) ‘Vibration of laminated-beams based on reference-plane formulation: Effect of end supports at different heights of the beam’, Engineering Structures. Elsevier, 159, pp. 245–251. doi: 10.1016/J.ENGSTRUCT.2018.01.004.

9. 9. Filippi, M., Carrera, E. and Valvano, S. (2018) ‘Analysis of multilayered structures embedding viscoelastic layers by higher-order, and zig-zag plate elements’, Composites Part B: Engineering. Elsevier, 154, pp. 77–89. doi: 10.1016/J.COMPOSITESB.2018.07.054.
10. 10. Gere, J. M. and Timoshenko, S. P. (1991) ‘Mechanics of Materials, 3rd Ed.’, in.
11. 11. Heyliger, P. R. and Reddy, J. N. (1988) ‘A higher order beam finite element for bending and vibration problems’, Journal of Sound and Vibration. Academic Press, 126(2), pp. 309–326. doi: 10.1016/0022-460X(88)90244-1.
12. 12. Iyengar, K. T. S. R. (2008) ‘APPLICATION OF MACLAURIN SERIES IN STRUCTURAL ANALYSIS’, Journal of the Indian Institute of Science, 8(3), pp. 879–887.
13. 13. Jena, S. K. et al. (2019) ‘Stability analysis of single-walled carbon nanotubes embedded in winkler foundation placed in a thermal environment considering the surface effect using a new refined beam theory’, Mechanics Based Design of Structures and Machines. Taylor & Francis, pp. 1–15. doi: 10.1080/15397734.2019.1698437.
14. 14. Jun, L. and Hongxing, H. (2009) ‘Variationally Consistent Higher-Order Analysis of Harmonic Vibrations of Laminated Beams’, Mechanics Based Design of Structures and Machines. Taylor & Francis, 37(3), pp. 299–326. doi: 10.1080/15397730902932608.
15. 15. Kant, T. and Gupta, A. (1988) ‘A finite element model for a higher-order shear-deformable beam theory’, Journal of Sound and Vibration. Academic Press, 125(2), pp. 193–202. doi: 10.1016/0022-460X(88)90278-7.
16. 16. Kim, N.-I. and Lee, J. (2015) ‘Refined Series Methodology for the Fully Coupled Thin-Walled Laminated Beams Considering Foundation Effects’, Mechanics Based Design of Structures and Machines. Taylor & Francis, 43(2), pp. 125–149. doi: 10.1080/15397734.2014.931811.
17. 17. Krishna Murty, A. V. (1985) ‘On the shear deformation theory for dynamic analysis of beams’, Journal of Sound and Vibration. Academic Press, 101(1), pp. 1–12. doi: 10.1016/S0022-460X(85)80033-X.
18. 18. Larbi, L. O. et al. (2013) ‘An Efficient Shear Deformation Beam Theory Based on Neutral Surface Position for Bending and Free Vibration of Functionally Graded Beams#’, Mechanics Based Design of Structures and Machines. Taylor & Francis, 41(4), pp. 421–433. doi: 10.1080/15397734.2013.763713.
19. 19. Levinson, M. (1981) ‘A new rectangular beam theory’, Journal of Sound and Vibration. Academic Press, 74(1), pp. 81–87. doi: 10.1016/0022-460X(81)90493-4.
20. 20. Levinson, M. (1985) ‘On Bickford’s consistent higher order beam theory’, Mechanics Research Communications. Pergamon, 12(1), pp. 1–9. doi: 10.1016/0093-6413(85)90027-8.
21. 21. Radice, J. J. (2012) ‘On the effect of local boundary condition details on the natural frequencies of simply-supported beams : Eccentric pin supports’, Mechanics Research Communications. Elsevier Ltd., 39(1), pp. 1–8. doi: 10.1016/j.mechrescom.2011.08.007.
22. 22. Reddy, J. N. (1997) ‘On locking-free shear deformable beam finite elements’, Computer Methods in Applied Mechanics and Engineering. North-Holland, 149(1–4), pp. 113–132. doi: 10.1016/S0045-7825(97)00075-3.
23. 23. Rehfield, L. W. and Murthy, P. L. N. (1982) ‘Toward a new engineering theory of bending - Fundamentals’, AIAA Journal. American Institute of Aeronautics and Astronautics, 20(5), pp. 693–699. doi: 10.2514/3.7938.
24. 24. Stephen, N. G. and Levinson, M. (1979) ‘A second order beam theory’, Journal of Sound and Vibration. Academic Press, 67(3), pp. 293–305. doi: 10.1016/0022-460X(79)90537-6.
25. 25. Timoshenko, S. P. (1923) ‘On the correction for shear of differential equation for transverse vibration of prismatic bars.’, Philosphical Magazine, 6(41), pp. 744–746.
26. 26. Türker, H. T. (2022) A modified beam theory for bending of eccentrically supported beams, Mechanics Based Design of Structures and Machines, 50:2, 576-587, DOI: 10.1080/15397734.2020.1738246
27. 27. Wang, C. M. et al. (2017) ‘Critical examination of midplane and neutral plane formulations for vibration analysis of FGM beams’, Engineering Structures. Elsevier, 130, pp. 275–281. doi: 10.1016/J.ENGSTRUCT.2016.10.051.
28. 28. Wang, C. M., Reddy, J. N. and Lee, K. H. (2000) Shear deformable beams and plates : relationships with classical solutions. Elsevier.
29. 29. Zhang, D.-G. and Zhou, Y.-H. (2008) ‘A theoretical analysis of FGM thin plates based on physical neutral surface’, Computational Materials Science. Elsevier, 44(2), pp. 716–720. doi: 10.1016/J.COMMATSCI.2008.05.016.