Novel Nonlinear Dynamic Beam–Foundation Interaction Model
Publication: Journal of Engineering Mechanics
Volume 147, Issue 4
Abstract
A nonlinear dynamic framework for the analysis of beam–foundation interaction is developed. The extended Hamilton’s principle is used to analyze the response of Euler–Bernoulli beams vibrating on nonlinear continuums (e.g., soils) and subjected to vibrating and moving loads. The continuum beneath the beam is characterized by a nonlinear elastic constitutive relationship that connects the secant shear modulus to the induced strain. The novel feature of the analysis is that the nonlinear compression and shear parameters and of the continuum (i.e., foundation) do not have to be assumed a priori, as is required in conventional beams on foundation analysis, and are obtained as part of the solution. In fact, it is shown that these parameters are not constant but change with time and depend on the beam–foundation interaction. Another novel feature of the analysis is that the mass of the foundation participating in the vibration is obtained as part of the solution and does not have to be assumed a priori. Thus, the analysis rigorously takes into account the nonlinear beam–foundation (soil–structure) interaction within a dynamic time-integration framework. The developed framework is as accurate as and about 50% faster than conventional nonlinear dynamic finite element analysis. Inputs to the analysis can be given in the form of a text file without any requirement of numerical mesh generation, making the approach rather user friendly. The characteristics of the developed nonlinear dynamic foundation model are illustrated through examples of moving and vibrating loads.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request as follows:
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Code used to generate some of the results in this paper.
References
Ai, Z. Y., Z. X. Li, and Y. C. Cheng. 2014. “BEM analysis of elastic foundation beams on multilayered isotropic soils.” Soils Found. 54 (4): 667–674. https://doi.org/10.1016/j.sandf.2014.06.008.
Ai, Z. Y., W. J. Liu, and B. K. Shi. 2018. “Quasi-static interaction between non-uniform beams and anisotropic permeable saturated multilayered soils with elastic superstrata.” Appl. Math. Modell. 53 (Jan): 400–418. https://doi.org/10.1016/j.apm.2017.08.020.
Aköz, A. Y., and F. Kadioğlu. 1996. “The mixed finite element solution of circular beam on elastic foundation.” Comput. Struct. 60 (4): 643–651. https://doi.org/10.1016/0045-7949(95)00418-1.
Antony, S. J., and Z. K. Jahanger. 2020. “Local scale displacement fields in grains–structure interactions under cyclic loading: Experiments and simulations.” Geotech. Geol. Eng. 38 (2): 1277–1294. https://doi.org/10.1007/s10706-019-01088-5.
Banerjee, P. K., and R. Butterfield. 1981. Vol. 17 of Boundary element methods in engineering science, 578. London: McGraw-Hill.
Bathe, K. J. 1996. Finite element procedures, 1037. Upper Saddle River, NJ: Prentice Hall.
Benz, T., P. A. Vermeer, and R. Schwab. 2009. “A small-strain overlay model.” Int. J. Numer. Anal. Methods Geomech. 33 (1): 25–44. https://doi.org/10.1002/nag.701.
Birkhoff, G. D. 1960. Vol. 9 of Dynamical systems. New York: American Mathematical Society.
Chopra, A. K. 1995. Dynamics of structures: Theory and applications to earthquake engineering. Upper Saddle River, NJ: Prentice Hall.
Darendeli, M. B. 2001. “Development of a new family of normalized modulus reduction and material damping curves.” Ph.D. dissertation, Dept. of Civil, Architectural and Environmental Engineering, Univ. of Texas at Austin.
Dobry, R., R. S. Ladd, F. Y. Yokel, R. H. Chung, and D. Powell. 1982. Prediction of pore pressure build-up and liquefaction of sand during earthquakes by the cyclic strain method. Washington, DC: National Bureau of Standards.
Duncan, J. M., and C. Y. Chang. 1970. “Nonlinear analysis of stress and strain in soils.” J. Soil Mech. Found. Div. 96 (5): 1629–1653.
Elhuni, H., and D. Basu. 2019. “Dynamic soil structure interaction model for beams on viscoelastic foundations subjected to oscillatory and moving loads.” Comput. Geotech. 115 (Nov): 103157. https://doi.org/10.1016/j.compgeo.2019.103157.
Fahey, M., and J. P. Carter. 1993. “A finite element study of the pressuremeter test in sand using a nonlinear elastic plastic model.” Can. Geotech. J. 30 (2): 348–362. https://doi.org/10.1139/t93-029.
Filonenko-Borodich, M. M. 1945. A very simple model of an elastic foundation capable of spreading the load. Moscow, Russia: Sbornik Moskovkovo Elektro Instituta.
Haldar, S., and D. Basu. 2016. “Analysis of beams on heterogeneous and nonlinear soil.” Int. J. Geomech. 16 (4): 04016004. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000599.
Hardin, B. O., and V. P. Drnevich. 1972. “Shear modulus and damping in soils: Design equations and curves.” J. Soil Mech. Found. Div. 98 (7): 667–692.
Ishibashi, I., and X. Zhang. 1993. “Unified dynamic shear moduli and damping ratios of sand and clay.” Soils Found. 33 (1): 182–191. https://doi.org/10.3208/sandf1972.33.182.
Jorge, P. C., F. M. F. Simões, and A. P. Da Costa. 2015. “Dynamics of beams on non-uniform nonlinear foundations subjected to moving loads.” Comput. Struct. 148 (Feb): 26–34. https://doi.org/10.1016/j.compstruc.2014.11.002.
Kargarnovin, M. H., D. Younesian, D. J. Thompson, and C. J. C. Jones. 2005. “Response of beams on nonlinear viscoelastic foundations to harmonic moving loads.” Comput. Struct. 83 (23–24): 1865–1877. https://doi.org/10.1016/j.compstruc.2005.03.003.
Kerr, A. D. 1964. “Elastic and viscoelastic foundation models.” J. Appl. Mech. 31 (3): 491–498. https://doi.org/10.1115/1.3629667.
Li, X., F. Xu, and Z. Zhang. 2017. “Symplectic eigenvalue analysis method for bending of beams resting on two-parameter elastic foundations.” J. Eng. Mech. 143 (9): 04017098. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001315.
Li, X., F. Xu, and Z. Zhang. 2018. “Symplectic method for natural modes of beams resting on elastic foundations.” J. Eng. Mech. 144 (4): 04018009. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001427.
Liang, R. Y., and J. X. Zhu. 1995. “Dynamic analysis of infinite beam on modified Vlasov subgrade.” J. Transp. Eng. 121 (5): 434–442. https://doi.org/10.1061/(ASCE)0733-947X(1995)121:5(434).
Limkatanyu, S., W. Prachasaree, N. Damrongwiriyanupap, M. Kwon, and W. Jung. 2013. “Exact stiffness for beams on Kerr-Type foundation: The virtual force approach.” J. Appl. Math. 2013: 134–146. https://doi.org/10.1155/2013/626287.
Liu, Q., and J. Ma. 2013. “Analytical model for beams on elastic foundations considering the coupling of horizontal and vertical displacements.” J. Eng. Mech. 139 (12): 1757–1768. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000635.
Lombardi, D., S. Bhattacharya, and D. M. Wood. 2013. “Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil.” Soil Dyn. Earthquake Eng. 49 (Jun): 165–180. https://doi.org/10.1016/j.soildyn.2013.01.015.
Ma, J., F. Liu, M. Nie, and J. Wang. 2018. “Nonlinear free vibration of a beam on Winkler foundation with consideration of soil mass motion of finite depth.” Nonlinear Dyn. 92 (2): 429–441. https://doi.org/10.1007/s11071-018-4066-8.
Mohanty, S. C., R. R. Dash, and T. Rout. 2011. “Parametric instability of a functionally graded Timoshenko beam on Winkler’s elastic foundation.” Nucl. Eng. Des. 241 (8): 2698–2715. https://doi.org/10.1016/j.nucengdes.2011.05.040.
Pasternak, P. L. 1954. On a new method of analysis of an elastic foundation by means of two foundation constants. Moscow: Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture.
Patil, V. A., V. A. Sawant, and K. Deb. 2013. “3D finite-element dynamic analysis of rigid pavement using infinite elements.” Int. J. Geomech. 13 (5): 533–544. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000255.
Rayleigh, J. W. S. B. 1896. Vol. 2 of The theory of sound. London: Macmillan.
Reissner, E. 1958. “Symmetric bending of shallow shells of revolution.” J. Math. Mech. 7 (2): 121–140.
Rice, J. R., and R. F. Boisvert. 1996. “From scientific software libraries to problem-solving environments.” IEEE Comput. Sci. Eng. 3 (3): 44–53. https://doi.org/10.1109/99.537091.
Sapountzakis, E. J., and A. E. Kampitsis. 2010. “Nonlinear dynamic analysis of Timoshenko beam-columns partially supported on tensionless Winkler foundation.” Comput. Struct. 88 (21–22): 1206–1219. https://doi.org/10.1016/j.compstruc.2010.06.010.
Sapountzakis, E. J., and A. E. Kampitsis. 2011. “Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads.” J. Sound Vib. 330 (22): 5410–5426. https://doi.org/10.1016/j.jsv.2011.06.009.
Sun, H., Y. Yang, L. Shi, and X. Geng. 2018. “The equivalent stiffness of a saturated poroelastic halfspace interacting with an infinite beam under a moving point load.” Soil Dyn. Earthq. Eng. 107 (Apr): 83–95. https://doi.org/10.1016/j.soildyn.2018.01.022.
Tsiatas, G. C. 2010. “Nonlinear analysis of non-uniform beams on nonlinear elastic foundation.” Acta Mech. 209 (1–2): 141. https://doi.org/10.1007/s00707-009-0174-3.
Vallabhan, C. G., and Y. C. Das. 1991. “A refined model for beams on elastic foundations.” Int. J. Solids Struct. 27 (5): 629–637. https://doi.org/10.1016/0020-7683(91)90217-4.
Vardanega, P. J., and M. D. Bolton. 2013. “Stiffness of clays and silts: Normalizing shear modulus and shear strain.” J. Geotech. Geoenviron. Eng. 139 (9): 1575–1589. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000887.
Vlasov, V. Z., and N. N. Leontiev. 1966. Beams, plates, and shells on elastic foundations, translated from Russian by Israel program for scientific translations. Washington, DC: National Aeronautics and Space Administration.
Vucetic, M., and R. Dobry. 1991. “Effect of soil plasticity on cyclic response.” J. Geotech. Eng. 117 (1): 89–107. https://doi.org/10.1061/(ASCE)0733-9410(1991)117:1(89).
Winkler, E. 1867. Theory of elasticity and strength. Prague, Czech Republic: Dominicus Prague.
Worku, A. 2012. “Calibrated analytical formulas for foundation model parameters.” Int. J. Geomech. 13 (4): 340–347. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000214.
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Published In
Journal of Engineering Mechanics
Volume 147 • Issue 4 • April 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Mar 6, 2020
Accepted: Dec 5, 2020
Published online: Jan 31, 2021
Published in print: Apr 1, 2021
Discussion open until: Jun 30, 2021
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