Nonlinear Finite Element Modeling of Deep Beam Resting on Linear and Nonlinear Random Soil

Authors: M. Seguini, D. Nedjar

Abstract:

An accuracy nonlinear analysis of a deep beam resting on elastic perfectly plastic soil is carried out in this study. In fact, a nonlinear finite element modeling for large deflection and moderate rotation of Euler-Bernoulli beam resting on linear and nonlinear random soil is investigated. The geometric nonlinear analysis of the beam is based on the theory of von Kàrmàn, where the Newton-Raphson incremental iteration method is implemented in a Matlab code to solve the nonlinear equation of the soil-beam interaction system. However, two analyses (deterministic and probabilistic) are proposed to verify the accuracy and the efficiency of the proposed model where the theory of the local average based on the Monte Carlo approach is used to analyze the effect of the spatial variability of the soil properties on the nonlinear beam response. The effect of six main parameters are investigated: the external load, the length of a beam, the coefficient of subgrade reaction of the soil, the Young’s modulus of the beam, the coefficient of variation and the correlation length of the soil’s coefficient of subgrade reaction. A comparison between the beam resting on linear and nonlinear soil models is presented for different beam’s length and external load. Numerical results have been obtained for the combination of the geometric nonlinearity of beam and material nonlinearity of random soil. This comparison highlighted the need of including the material nonlinearity and spatial variability of the soil in the geometric nonlinear analysis, when the beam undergoes large deflections.

Keywords: Finite element method, geometric nonlinearity, material nonlinearity, soil-structure interaction, spatial variability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1339099

References:

[1] Hetenyi, Beams on elastic foundations. Ann Arbor, MI: University of Michigan Press, USA, 1961.
[2] S. Timoshenko, Strength of Materials, Part II, Advanced Theory and Problems. 3rd ed., Princeton, NJ: Van Nostrand, USA, 1956.
[3] S. Motohiro, K. Shunji and M. Takashi, “Structural modeling of beams on elastic foundations with elasticity couplings,” Mechanics Research Communications, vol. 34, no. 5-6, pp. 451-459, 2007.
[4] C. Miranda and K. Nair, “Finite beams on elastic foundation,” ASCE Journal of Structure Division, vol. 92, no. ST2, Paper 4778, pp. 131-142, 1966.
[5] M. Eisenberger and J. Clastornik, “Beams on variable two-parameter elastic foundation,” Journal of Engineering Mechanics, vol. 113, no. 10, pp. 1454-1466, 1987.
[6] B. Y. Ting and E. F. Mockry, “Beam on elastic foundation finite elements,” Journal of Structural Engineering, vol. 110, no. 10, pp. 2324-2339, 1984.
[7] E. Winkler,” Die Lehre von der Elasticitaet und Festigkeit (The theory of elasticity and strength),” Dominicus: Prag, 1867.
[8] M. Eisenberger and D. Z. Yankelevsky, “Exact stiffness matrix for beams on elastic foundation,” Computers and Structures, vol. 21, no. 6, pp. 1355-1359, 1985.
[9] D. Z. Yankelevsky, M. Eisenberger and M. A. Adin, “Analysis of beams on nonlinear Winkler foundation,” Computers and Structures, vol. 31, no. 2, pp. 287-292, 1989.
[10] J. P. Beaufait and W. Hoadley, “Analysis of elastic beams on nonlinear foundations,” Computers and Structures, vol. 12, no. 5, pp. 669-676, 1980.
[11] C. W. Harden and T. C. Hutchinson, “Beam on nonlinear Winkler foundation modeling of shallow rocking-dominated footings,” Earthquake Spectra, vol. 25, no. 2, pp. 277-300, 2009.
[12] S. P. Sharma and S. Dasgupta, “The bending problem of axially constrained beams on nonlinear elastic foundations,” International Journal of Solids and Structures, vol. 11, pp. 853-889, 1975.
[13] T. M. Wang and L. W. Gagnon, “Vibrations of continuous Timoshenko beams on Winkler-Pasternak foundations,” Journal of Sound and Vibration, vol. 59, no. 2, pp. 211-220, 1978.
[14] F. Fırat Çalım, “Dynamic analysis of beams on viscoelastic foundation,” European Journal of Mechanics-A/Solids, vol. 28, no. 3, pp. 469-476, 2009.
[15] C. Bridge and N. Willis, “Steel catenary risers results and conclusions from large-scale simulations of seabed interactions,” Proceedings of the International Conference on Deep Offshore Technology, New Orleans, Louisiana, (2002).
[16] C. Bridge, K. Laver, E. Clukey and T. Evans, “Steel catenary riser touchdown point vertical interaction models,” Proceedings of the Conference on Offshore Technology, Houston, Texas, 2004.
[17] M. S. Hodder and B. W. Byrne, “3D experiments investigating the interaction of a model SCR with the seabed,” Applied Ocean Research, vol. 32, no. 2, pp. 146–157, 2010.
[18] K. J. Bathe, “Finite element procedures in engineering analysis,” Englewood Cliffs, NJ: Prentice-Hall, 1982.
[19] J. N. Reddy, “An introduction to nonlinear finite element analysis,” Oxford University Press, 2004.
[20] S. A. Hosseini Kordkheili and H. Bahai, “Non-linear finite element analysis of flexible risers in presence of buoyancy force and seabed interaction boundary condition,” Archive of Applied Mechanics, vol. 78, no. 10, pp. 765–774, 2008.
[21] S. A. Hosseini Kordkheili, H. Bahai and M. Mirtaheri, “An updated Lagrangian finite element formulation for large displacement dynamic analysis of three-dimensional flexible riser structures,” Ocean Engineering, vol. 38, no. 5-6, pp. 793-803, 2011.
[22] T. Horibe, “An analysis for large deflection problems of beams on elastic foundations by boundary integral equation method,” Transaction of Japan Society of Mechanical Engineers (JSME)-Part A, vol. 53, no. 487, pp. 622-629, 1987.
[23] T. S. Jang, H. S. Baek and J. K. Paik, “A new method for the nonlinear deflection analysis of an infinite beam resting on a nonlinear elastic foundation,” International Journal of Non-Linear Mechanics, vol. 46, no. 1, pp. 339–346, 2011.
[24] T. S. Jang, “A new semi-analytical approach to large deflections of Bernoulli–Euler-v. Karman beams on a linear elastic foundation: Nonlinear analysis of infinite beams,” International Journal of Mechanical Sciences, vol. 66, pp. 22–32, 2013.
[25] T. S. Jang, “A general method for analysing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation,” Acta Mechanica, vol. 225, no. 7, pp. 1967-1984, 2014.
[26] A. A. Al- Azzawi, H. Mahdy and O. Sh. Farhan, “Finite element analysis of deep beams on nonlinear elastic foundations,” Journal of the Serbian Society for Computational Mechanics, vol. 4, no. 2, pp. 13-42, 2010.
[27] D. M. Al-Talaqany, “Large Deflection Deep Beams on Elastic Foundations,” M.Sc.Thesis, Faculty of Engineering, Nahrain University of Baghdad, Iraq, 2007.
[28] A. A. Al- Azzawi and D. M. Theeban, “Large deflection of deep beams on Elastic Foundations,” Journal of the Serbian Society for Computational Mechanics, vol. 4 no. 1, pp. 88-101, 2010.
[29] D. V. Griffiths, J. Paiboon, J. Huang, G. A. Fenton, “Numerical analysis of beams on random elastic foundations,” In: Proceedings of the 9th international congress on numerical methods in engineering and scientific applications, CIMENICS, pp. 19–25, 2008.
[30] S. M. Elachachi, D. Breysse and L. Houy, “Longitudinal variability of soils and structural response of sewer networks,” Computers and Geotechnics, vol. 31, no. 8, pp. 625–641, 2004.
[31] E. VanMarcke, “Random fields: Analysis and synthesis,” Cambridge, MA: MIT Press, 1983.
[32] D. Nedjar, M. Bensafi, S. M. Elachachi, M. Hamane and D. Breysse, “Buried pipe response under seismic solicitation with soil–pipe interaction,” In Mestat (Ed.), NUMGE conference Paris: ENPC/ LCPC, pp. 1047–1053, 2002.
[33] D. Nedjar, M. Hamane, M. Bensafi, S. M. Elachachi and D. Breysse, “Seismic response analysis of pipes by a probabilistic approach,” Soil Dynamics and Earthquake Engineering, vol. 27, no. 2, pp. 111–115, 2007.
[34] S. M. Elachachi, D. Breysse and H. Benzeguir, “Soil spatial variability and structural reliability of buried networks subjected to earthquakes”, Computational Methods in Applied Sciences, vol. 22, pp. 111–127, 2011.
[35] S. M. Elachachi, D. Breysse and A. Denis, “Effect of soil spatial variability on reliability of rigid buried pipes,” Computers and Geotechnics, vol. 43, pp. 61–71, 2012.
[36] N. Kazi Tani, D. Nedjar and M. Hamane, “Non-linear analysis of the behaviour of buried structures in random media,” European Journal of Environmental and Civil Engineering, vol. 17, no. 9, pp. 791-801, 2013.
[37] G. A. Fenton and E. H. VanMarcke, “Simulation of random fields via local average subdivision,” Journal of Engineering Mechanics, vol. 116, no. 8, pp.733–1749, 1990.