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New energy functional for analysis of statically indeterminate axially loaded viscoelastic bars

Year 2023, Volume: 12 Issue: 1, 216 - 224, 15.01.2023
https://doi.org/10.28948/ngumuh.1187163

Abstract

In elastic bodies, stress is only a function of strain, while in viscoelastic bodies, stress depends on both strain and strain rate. By making various combinations of springs and dashpots with different material constants, it is possible to represent the mechanical behavior of materials such as high polymers, nylon fibers, concrete etc. In this study, which discusses the statically indeterminate axially loaded bar problem, whose mechanical behavior is represented using the Maxwell model, a solution technique that can be easily applied to even the most complex structural systems is proposed by using the total potential energy (TPE) theorem. The TPE expression in terms of the displacement of the nodes is obtained in the Laplace space. The solutions that minimize the TPE expression are the real displacements, and the Inverse Laplace transform method is applied to return to the time domain. The method has been tested on the sample problem and the results are presented. This method provides great convenience in obtaining the solution directly by following a few simple process steps, regardless of the change in the viscoelastic material model, the number of elements of the system and the type of loading.

References

  • T-M. Chen, The hybrid Laplace transform/finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. International Journal for Numerical Methods in Engineering, 38, 509-522, 1995. https://doi.org/ 10.1002/nme.1620380310.
  • G. Honig and U. Hirdes, A method for the numerical inversion of Laplace transform. Journal of Computational and Applied Mathematics, 10(1), 113-132,1984. https://doi.org/10.1016/0377-0427 (84)90075-X.
  • J.D. Mehl, R.N. Miles, Finite element modeling of the transient response of viscoelastic beams. Proceedings of the SPIE, pp. 306-311, 1995.
  • A.Y. Aköz, F. Kadıoğlu, The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. International Journal for Numerical Methods in Engineering, 44, 1909–1932, 1999.https://doi.org/10.1002/(SICI)10970207(19990430)44:12<1909::AID-NME573>3.0.CO;2-P.
  • F.S. Barbosa, M.C.R. Farage, A finite element model for sandwich viscoelastic beams: Experimental and numerical assessment. Journal of Sound and Vibration, 317, 91-111, 2008. https://doi.org/10.1016/ j.jsv.2008.03.013.
  • A. Pálfalvi, A comparison of finite element formulations for dynamics of viscoelastic beams. Finite Elements in Analysis and Design, 44,814- 818, 2008. https://doi.org/10.1016/j.finel.2008.06.009.
  • M. Enelund, G.A. Lesieutre, Time domain modeling of damping using anelastic displacement fields and fractional calculus. International Journal of Solids and Structures, 36 (29), 4447-4472, 1999. https://doi.org/10.1016/S0020-7683(98)00194-2.
  • F. Kpeky, H. Boudaoud, F. Abed-Meraim, E.M. Daya, Modeling of viscoelastic sandwich beams using solid-shell finite elements. Composite Structures, 133,105-116,2015.https://doi.org/10.1016/j.compstruct.2015.07.055.
  • Z. Huang, Xi. Wang, N. Wu, F. Chu, and J. Luo, A finite element model for the vibration analysis of sandwich beam with frequency-dependent viscoelastic material core. Materials, 12 (20), 3390, 2019. https://doi.org/10.3390/ma12203390.
  • M. Filippi, E. Carrera, Stress analyses of viscoelastic three-dimensional beam-like structures with low- and high-order one-dimensional finite elements. Meccanica, 56, 1475-1482, 2021. https://doi.org/10.1007/s11012-020-01191-5.
  • M. Arda, Vibration analysis of an axially loaded viscoelastic nanobeam. International Journal of Engineering and Applied Sciences, 10 (3), 252-263, 2018. https://doi.org/10.24107/ijeas.468769.
  • A. Shariati, D.W. Jung, H.M. Sedighi, K.K. Zur, M. Habibi, M. Safa, On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials, 13(7), 1707, 2020. https://doi.org/10.3390/ma13071707.
  • A. Ebrahimi-Mamaghani, A. Forooghi, H. Sarparast, A. Alibeigloo, M.I. Friswell, Vibration of viscoelastic axially graded beams with simultaneous axial and spinning motions under an axial load. Applied Mathematical Modelling, 90,131-150, 2021. https://doi.org/10.1016/j.apm.2020.08.041.
  • S. Liu, Y.Q. Tang, L. Chen, Multi-scale analysis and Galerkin verification for dynamic stability of axially translating viscoelastic Timoshenko beams. Applied Mathematical Modelling, 93, 885-897, 2021. https://doi.org/10.1016/j.apm.2020.12.039.
  • A. Mokhtari, H.R. Mirdamadi, Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: non-transforming spectral element analysis. Applied Mathematical Modelling, 56, 342-358, 2018. https://doi.org/10.1016/j.apm.2017.12.007.
  • M. Gürgöze, Parametric vibrations of a viscoelastic beam (Maxwell model) under steady axial load and transverse displacement excitation at one end. Journal of Sound and Vibration, 115(8), 329-338, 1987. https://doi.org/10.1016/0022-460X(87)90476-7.
  • U.S. Shirahatti, S.C. Sinha, Stability of perfect viscoelastic columns subjected to periodic axial loading, Proceedings of the ASME 1991 Design Technical Conferences. 13th Biennial Conference on Mechanical Vibration and Noise: Vibration Analysis -Analytical and Computational. Miami, Florida, USA, 225-231, 1991.
  • A. Manevich and Z. Kołakowski, Free and forced oscillations of Timoshenko beam made of viscoelastic material. Journal of Theoretical and Applied Mechanics, 49(1), 3-16, 2011.
  • J. Freundlich, Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load. Journal of Theoretical and Applied Mechanics, 54(4), 1433-1445, 2016. https://doi.org/10.15632/jtam-pl.54.4.1433.
  • A. H. Sofiyev, On the solution of the dynamic stability of heterogeneous orthotropic visco-elastic cylindrical shells. Composite Structures, 206, 124-130, 2018. https://doi.org/10.1016/j.compstruct.2018.08.027.
  • A. H. Sofiyev, Z. Zerin, N. Kuruoglu, Dynamic behavior of FGM viscoelastic plates resting on elastic foundations. Acta Mechanica, 231, 1-17, 2020. https://doi.org/10.1007/s00707-019-02502-y.
  • A. H. Sofiyev, About an approach to the determination of the critical time of viscoelastic functionally graded cylindrical shells. Composites Part B: Engineering, 156, 156-165, 2019. https://doi.org/10.1016/ j.compositesb.2018.08.073.
  • H.A. Zamani, M.M. Aghdam, M. Sadighi, Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory. Composite Structures, 182(15), 25-35, 2017. https://doi.org/10.1016/j.compstruct.2017.08.101.
  • G. Tekin, F, Kadıoğlu, Viscoelastic behavior of shear-deformable plates. International Journal of Applied Mechanics 9(6), 1750085 (23 pages), 2017. https://doi.org/10.1142/S1758825117500855.
  • D. Gutierrez-Lemini, Engineering Viscoelasticity, Springer, New York, 2014.

Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli

Year 2023, Volume: 12 Issue: 1, 216 - 224, 15.01.2023
https://doi.org/10.28948/ngumuh.1187163

Abstract

Elastik cisimlerde gerilme sadece şekil değiştirmenin bir fonksiyonudur, viskoelastik cisimlerde ise gerilme hem şekil değiştirmeye hem de şekil değiştirme hızına bağlıdır. Maddesel sabitleri farklı olan yayların ve sönüm kutularının çeşitli kombinasyonları yapılarak, yüksek polimerler, naylon lifler, beton vb. malzemelerin mekanik davranışlarını temsil etme olanağı vardır. Maxwell modeli kullanılarak mekanik davranışı temsil edilen statikçe belirsiz eksenel yüklü çubuk probleminin ele alındığı bu çalışmada, toplam potansiyel enerji (TPE) teoremi kullanılarak en karmaşık yapı sistemlerine bile kolaylıkla uygulanabilecek bir çözüm yolu önerilmiştir. Düğüm noktalarının yer değiştirmeleri cinsinden bulunan TPE ifadesi Laplace uzayında elde edilmiştir. TPE ifadesini minimum yapan çözümler gerçek yer değiştirmeler olup, Laplace uzayında elde edilen çözümlerden zaman uzayına geçmek için Ters Laplace dönüşümü yöntemi uygulanmıştır. Yöntem örnek problem üzerinde test edilmiş ve sonuçlar sunulmuştur. Bu yöntem, viskoelastik malzeme modelinin, sistemi oluşturan eleman sayısının ve yükleme tipinin değişmesinden bağımsız olarak birkaç basit işlem adımının takibi ile doğrudan çözüme ulaşmada büyük kolaylık sağlar.

References

  • T-M. Chen, The hybrid Laplace transform/finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. International Journal for Numerical Methods in Engineering, 38, 509-522, 1995. https://doi.org/ 10.1002/nme.1620380310.
  • G. Honig and U. Hirdes, A method for the numerical inversion of Laplace transform. Journal of Computational and Applied Mathematics, 10(1), 113-132,1984. https://doi.org/10.1016/0377-0427 (84)90075-X.
  • J.D. Mehl, R.N. Miles, Finite element modeling of the transient response of viscoelastic beams. Proceedings of the SPIE, pp. 306-311, 1995.
  • A.Y. Aköz, F. Kadıoğlu, The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. International Journal for Numerical Methods in Engineering, 44, 1909–1932, 1999.https://doi.org/10.1002/(SICI)10970207(19990430)44:12<1909::AID-NME573>3.0.CO;2-P.
  • F.S. Barbosa, M.C.R. Farage, A finite element model for sandwich viscoelastic beams: Experimental and numerical assessment. Journal of Sound and Vibration, 317, 91-111, 2008. https://doi.org/10.1016/ j.jsv.2008.03.013.
  • A. Pálfalvi, A comparison of finite element formulations for dynamics of viscoelastic beams. Finite Elements in Analysis and Design, 44,814- 818, 2008. https://doi.org/10.1016/j.finel.2008.06.009.
  • M. Enelund, G.A. Lesieutre, Time domain modeling of damping using anelastic displacement fields and fractional calculus. International Journal of Solids and Structures, 36 (29), 4447-4472, 1999. https://doi.org/10.1016/S0020-7683(98)00194-2.
  • F. Kpeky, H. Boudaoud, F. Abed-Meraim, E.M. Daya, Modeling of viscoelastic sandwich beams using solid-shell finite elements. Composite Structures, 133,105-116,2015.https://doi.org/10.1016/j.compstruct.2015.07.055.
  • Z. Huang, Xi. Wang, N. Wu, F. Chu, and J. Luo, A finite element model for the vibration analysis of sandwich beam with frequency-dependent viscoelastic material core. Materials, 12 (20), 3390, 2019. https://doi.org/10.3390/ma12203390.
  • M. Filippi, E. Carrera, Stress analyses of viscoelastic three-dimensional beam-like structures with low- and high-order one-dimensional finite elements. Meccanica, 56, 1475-1482, 2021. https://doi.org/10.1007/s11012-020-01191-5.
  • M. Arda, Vibration analysis of an axially loaded viscoelastic nanobeam. International Journal of Engineering and Applied Sciences, 10 (3), 252-263, 2018. https://doi.org/10.24107/ijeas.468769.
  • A. Shariati, D.W. Jung, H.M. Sedighi, K.K. Zur, M. Habibi, M. Safa, On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials, 13(7), 1707, 2020. https://doi.org/10.3390/ma13071707.
  • A. Ebrahimi-Mamaghani, A. Forooghi, H. Sarparast, A. Alibeigloo, M.I. Friswell, Vibration of viscoelastic axially graded beams with simultaneous axial and spinning motions under an axial load. Applied Mathematical Modelling, 90,131-150, 2021. https://doi.org/10.1016/j.apm.2020.08.041.
  • S. Liu, Y.Q. Tang, L. Chen, Multi-scale analysis and Galerkin verification for dynamic stability of axially translating viscoelastic Timoshenko beams. Applied Mathematical Modelling, 93, 885-897, 2021. https://doi.org/10.1016/j.apm.2020.12.039.
  • A. Mokhtari, H.R. Mirdamadi, Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: non-transforming spectral element analysis. Applied Mathematical Modelling, 56, 342-358, 2018. https://doi.org/10.1016/j.apm.2017.12.007.
  • M. Gürgöze, Parametric vibrations of a viscoelastic beam (Maxwell model) under steady axial load and transverse displacement excitation at one end. Journal of Sound and Vibration, 115(8), 329-338, 1987. https://doi.org/10.1016/0022-460X(87)90476-7.
  • U.S. Shirahatti, S.C. Sinha, Stability of perfect viscoelastic columns subjected to periodic axial loading, Proceedings of the ASME 1991 Design Technical Conferences. 13th Biennial Conference on Mechanical Vibration and Noise: Vibration Analysis -Analytical and Computational. Miami, Florida, USA, 225-231, 1991.
  • A. Manevich and Z. Kołakowski, Free and forced oscillations of Timoshenko beam made of viscoelastic material. Journal of Theoretical and Applied Mechanics, 49(1), 3-16, 2011.
  • J. Freundlich, Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load. Journal of Theoretical and Applied Mechanics, 54(4), 1433-1445, 2016. https://doi.org/10.15632/jtam-pl.54.4.1433.
  • A. H. Sofiyev, On the solution of the dynamic stability of heterogeneous orthotropic visco-elastic cylindrical shells. Composite Structures, 206, 124-130, 2018. https://doi.org/10.1016/j.compstruct.2018.08.027.
  • A. H. Sofiyev, Z. Zerin, N. Kuruoglu, Dynamic behavior of FGM viscoelastic plates resting on elastic foundations. Acta Mechanica, 231, 1-17, 2020. https://doi.org/10.1007/s00707-019-02502-y.
  • A. H. Sofiyev, About an approach to the determination of the critical time of viscoelastic functionally graded cylindrical shells. Composites Part B: Engineering, 156, 156-165, 2019. https://doi.org/10.1016/ j.compositesb.2018.08.073.
  • H.A. Zamani, M.M. Aghdam, M. Sadighi, Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory. Composite Structures, 182(15), 25-35, 2017. https://doi.org/10.1016/j.compstruct.2017.08.101.
  • G. Tekin, F, Kadıoğlu, Viscoelastic behavior of shear-deformable plates. International Journal of Applied Mechanics 9(6), 1750085 (23 pages), 2017. https://doi.org/10.1142/S1758825117500855.
  • D. Gutierrez-Lemini, Engineering Viscoelasticity, Springer, New York, 2014.
There are 25 citations in total.

Details

Primary Language Turkish
Subjects Civil Engineering
Journal Section Civil Engineering
Authors

Gülçin Tekin 0000-0003-0207-4305

Fethi Kadıoğlu 0000-0001-7049-1704

Publication Date January 15, 2023
Submission Date October 10, 2022
Acceptance Date November 7, 2022
Published in Issue Year 2023 Volume: 12 Issue: 1

Cite

APA Tekin, G., & Kadıoğlu, F. (2023). Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 12(1), 216-224. https://doi.org/10.28948/ngumuh.1187163
AMA Tekin G, Kadıoğlu F. Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli. NOHU J. Eng. Sci. January 2023;12(1):216-224. doi:10.28948/ngumuh.1187163
Chicago Tekin, Gülçin, and Fethi Kadıoğlu. “Hiperstatik Eksenel yüklü Viskoelastik çubukların Analizi için Yeni Enerji Fonksiyoneli”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 12, no. 1 (January 2023): 216-24. https://doi.org/10.28948/ngumuh.1187163.
EndNote Tekin G, Kadıoğlu F (January 1, 2023) Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 12 1 216–224.
IEEE G. Tekin and F. Kadıoğlu, “Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli”, NOHU J. Eng. Sci., vol. 12, no. 1, pp. 216–224, 2023, doi: 10.28948/ngumuh.1187163.
ISNAD Tekin, Gülçin - Kadıoğlu, Fethi. “Hiperstatik Eksenel yüklü Viskoelastik çubukların Analizi için Yeni Enerji Fonksiyoneli”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi 12/1 (January 2023), 216-224. https://doi.org/10.28948/ngumuh.1187163.
JAMA Tekin G, Kadıoğlu F. Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli. NOHU J. Eng. Sci. 2023;12:216–224.
MLA Tekin, Gülçin and Fethi Kadıoğlu. “Hiperstatik Eksenel yüklü Viskoelastik çubukların Analizi için Yeni Enerji Fonksiyoneli”. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, vol. 12, no. 1, 2023, pp. 216-24, doi:10.28948/ngumuh.1187163.
Vancouver Tekin G, Kadıoğlu F. Hiperstatik eksenel yüklü viskoelastik çubukların analizi için yeni enerji fonksiyoneli. NOHU J. Eng. Sci. 2023;12(1):216-24.

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