Analysis of beam-column elements on non-homogeneous soil using the differential transformation method
DOI:
https://doi.org/10.17533/udea.redin.20210218Keywords:
Beam-column element, mathematical analysis, non-homogeneous soil, soil mechanicsAbstract
This paper describes an analytical approach to conduct an analysis of beam-column elements with generalized end-boundary conditions on a homogeneous or non-homogeneous Pasternak elastic foundation. The mathematical formulation utilized herein is that presented by the senior author in a recent work. The differential equation (DE) governing the behavior of the beam-column element is solved using the differential transformation method (DTM). The DTM offers practical advantages over other conventional approaches when solving the proposed structural model. The proposed formulation provides the flexibility to account for i) combined lateral and axial load at the ends of the element, ii) homogeneous or non-homogeneous soil, iii) Pasternak elastic foundation, and iv) an external arbitrary transverse load acting on the element. The effects of various slenderness ratios, pile-soil stiffness ratios, and classical and semirigid boundary conditions can be easily studied with the proposed formulation. Examples are presented to validate the accuracy of the model and its applicability over a wide range of analyses.
Downloads
References
E. Winkler, Die Lehre von der Elastizitaet und Festigkeit. Prague: Dominicus, 1867.
H. Zimmermann, Die Berechnung des Eisenbahnoberbaues, 2nd ed. Berlin: Ernst & Korn, 1888.
A. Dodge, “Influence Functions for Beams on Elastic Foundations,” Journal of the Structural Division, vol. 90, no. 4, pp. 63–102, 1964.
C. Miranda and K. Nair, “Finite Beams on Elastic Foundations,” Journal of the Structural Division, vol. 92, no. 2, pp. 131–142, 1966.
M. Hetenyi, “Beams on elastic foundation; theory with applications in the fields of civil and mechanical engineering,” pp. ix, 255 p., 1946.
K. Terzaghi, “Evaluation of coefficients of subgrade reaction,” Geotechnique, vol. 5, no. 4, pp. 41–50, 1955.
A.-M. L. G., Z.-M. D. G., and A.-O. J. Darío, “Reut and beck columns: effects of end gravity force, translation and rotational inertias,” Revista Facultad de Ingeniería Universidad de Antioquia, no. 65, pp. 191–200, 2013. [Online]. Available: https://revistas.udea.edu.co/ index.php/ingenieria/article/view/14229
J. D. Aristizabal-Ochoa, “Stability of slender columns on an elastic foundation with generalized end conditions,” Ingeniería e Investigación, vol. 33, no. 3, pp. 34–40, 2013. [Online]. Available: https://revistas.unal.edu.co/index.php/ingeinv/article/view/41041
P. L. Pasternak, On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants. Moscow: Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvui Arkhitekture, 1954.
A. D. Kerr, “Elastic and Viscoelastic Foundation Models,” Journal of Applied Mechanics, vol. 31, no. 3, pp. 491–498, 9 1964. [Online]. Available: https://asmedigitalcollection.asme.org/appliedmechanics/article/ 31/3/491/386992/Elastic-and-Viscoelastic-Foundation-Models
V. Z. Vlasov and N. N. Leontev, Beams, plates and shells on elastic foundations. Jerusalem: Israel Program for Scientific Translations, 1966.
I. H. Abdel-Halim Hassan, “Different applications for the differential transformation in the differential equations,” Applied Mathematics and Computation, vol. 129, no. 2-3, pp. 183–201, 2002. [Online]. Available: https://doi.org/10.1016/S0096-3003(01)00037-6
I. H. Abdel-Halim Hassan, “Application to differential transformation method for solving systems of differential equations,” Applied Mathematical Modelling, vol. 32, no. 12, pp. 2552–2559, 2008. [Online]. Available: https://doi.org/10.1016/j.apm.2007.09.025
C.-K. Chen and S.-H. Ho, “Application of differential transformation to eigenvalue problems,” Applied Mathematics and Computation, vol. 79, no. 2-3, pp. 173–188, 10 1996. [Online]. Available: https://doi.org/10.1016/0096-3003(95)00253-7
J. K. Zhou, “Differential transformation and its applications for electrical circuits,” Wuhan, China, pp. 1279–1289, 1986.
B. Bozyigit and Y. Yesilce, “Dynamic stiffness approach and differential transformation for free vibration analysis of a moving reddy-bickford beam,” Structural Engineering and Mechanics, vol. 58, no. 5, pp. 847–868, 2016. [Online]. Available: https://doi.org/10. 12989/sem.2016.58.5.847
B. Bozyigit, Y. Yesilce, and S. Catal, “Free vibrations of axial-loaded beams resting on viscoelastic foundation using adomian decomposition method and differential transformation,” Engineering science and technology, an international journal, vol. 21, no. 6, pp. 1181–1193, 2018.
S. Çatal, “Buckling analysis of partially embedded pile in elastic soil using differential transform method,” Structural Engineering and Mechanics, vol. 24, no. 2, pp. 247–268, 9 2006. [Online]. Available: https://doi.org/10.12989/sem.2006.24.2.247
S. Çatal, “Solution of free vibration equations of beam on elastic soil by using differential transform method,” Applied Mathematical Modelling, vol. 32, no. 9, pp. 1744–1757, 2008. [Online]. Available: https://doi.org/10.1016/j.apm.2007.06.010
Y. Yesilce, “Differential transform method for free vibration analysis of a moving beam,” Structural Engineering and Mechanics, vol. 35, no. 5, pp. 645–658, 7 2010. [Online]. Available: https://doi.org/10. 12989/sem.2010.35.5.645
M. Balkaya, M. O. Kaya, and A. Saglamer, “Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method,” Archive of Applied Mechanics, vol. 79, no. 2, pp. 135–146, 2009. [Online]. Available: https: //doi.org/10.1007/s00419-008-0214-9
B. Bozyigit, Y. Yesilce, and S. Catal, “Differential transform method and adomian decomposition method for free vibration analysis of fluid conveying timoshenko pipeline,” Structural Engineering and Mechanics, vol. 62, no. 1, pp. 65–77, 2017. [Online]. Available: https://doi.org/10.12989/sem.2017.62.1.065
J. D. Aristizabal-Ochoa, “First- and second-order stiffness matrices and load vector of beam-columns with semirigid connections,” Journal of Structural Engineering, vol. 123, no. 5, pp. 669–678, 1997. [Online]. Available: https://doi.org/10.1061/(ASCE)0733-9445(1997) 123:5(669)
C. A. Vega-Posada, A. P. Gallant, and M. Areiza-Hurtado, “Simple approach for analysis of beam-column elements on homogeneous and non-homogeneous elastic soil,” Engineering Structures, vol. 221, p. 111110, 2020. [Online]. Available: https://doi.org/10.1016/j. engstruct.2020.111110
S. v20.0.0., Computers & Structures.INC, Walnut Creek, CA.
L. G. Arboleda-Monsalve, D. G. Zapata-Medina, and J. D. Aristizabal-Ochoa, “Timoshenko beam-column with generalized end conditions on elastic foundation: Dynamic-stiffness matrix and load vector,” Journal of Sound and Vibration, vol. 310, no. 4-5, pp. 1057–1079, 2008. [Online]. Available: https: //doi.org/10.1016/j.jsv.2007.08.014
A. F. Ramirez-Henao and J. Paul Smith-Pardo, “Elastic stability of pile-supported wharves and piers,” Engineering Structures, vol. 97, pp. 140–151, 2015. [Online]. Available: https://doi.org/10.1016/j. engstruct.2015.04.007
P. K. Banerjee and T. G. Davies, “The Behaviour of Axially and Laterally Loaded Sigle Piles Embedded In Non-Homogeneous Soils,” Geotechnique, vol. 30, no. 1, pp. 88–92, 1980. [Online]. Available: https://doi.org/10.1680/geot.1980.30.1.88
M. F. Randolph, “The response of flexible piles to lateral loading,” Geotechnique, vol. 31, no. 2, pp. 247–259, 1981. [Online]. Available: https://doi.org/10.1680/geot.1981.31.2.247
W. Higgins, C. Vasquez, D. Basu, and D. V. Griffiths, “Elastic solutions for laterally loaded piles,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 139, no. 7, pp. 1096–1103, 2013. [Online]. Available: https://doi.org/10.1061/(ASCE)GT.1943-5606. 0000828
D. Basu and R. Salgado, “Method of initial parameters for laterally loaded piles embedded in layered soils,” Geomechanics and Geoengineering, vol. 2, no. 4, pp. 281–294, 2007. [Online]. Available: https://doi.org/10.1080/17486020701678869
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Revista Facultad de Ingeniería Universidad de Antioquia
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Revista Facultad de Ingeniería, Universidad de Antioquia is licensed under the Creative Commons Attribution BY-NC-SA 4.0 license. https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en
You are free to:
Share — copy and redistribute the material in any medium or format
Adapt — remix, transform, and build upon the material
Under the following terms:
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
NonCommercial — You may not use the material for commercial purposes.
ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.
The material published in the journal can be distributed, copied and exhibited by third parties if the respective credits are given to the journal. No commercial benefit can be obtained and derivative works must be under the same license terms as the original work.
Twitter