Meshless Analysis of a box culvert resting on Modified Vlasov Foundation

Document Type : Research


1 Civil Engineer, Department of Civil Engineering, ENSPY, University of Yaounde I, Cameroon.

2 Lecturer, Department of MSP and Civil Engineering, ENSPY, Uni-versity of Yaounde I, Cameroon.

3 Assistant Professor, Department of Civil Engineering, ENSPY, Uni-versity of Yaounde I, Cameroon.


In this paper, we compare computational design of box culverts using Kleinlogel’s formulas and three-dimensional models based on Winkler and Modified Vlasov Foundation (MVF) by Radial Point Interpolation method (RPIM) . In order to extend the RPIM to the study of box culverts, a method of decomposition by subdomain and a creation of a fictitious rotation is made. Similar results are observed between the RPIM_Winkler and STAAD PRO (FEM software) with fewer nodes in the case study. The Kleinlogel’s formulas or the Winkler model gives a maximal increase in stresses at the middle of the raft of 5%, and a 12 times increased displacement. Finally, it emerges that for the same type of soil and a single structure, the reaction and shear modulus of the soil are highly dependent on the distribution of loads on the structure.


[1] Y. M. Ghugal and R P Shimpi (2002), A review of re-fined shear deformation theories of isotropic and aniso-tropic laminated plates, J. Reinf. Plast. Compos., Vol. 21, pp. 775-813  
[2] R.A. Shetty, Dushyanthkumar G.L., Deepak S.A. (2018), Classical and Refined Beam and Plate Theories: A Brief Technical Review, Sahyadri International Journal of Research; 4(2), pp27-32  
[3] E. Reissner (1945), The effect of transverse shear de-formation on the bending of elastic plates, J. Appl. Mech., Vol. 12, pp69-77  
[4] R. D. Mindlin (1951), Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech., Vol. 18, pp. 31-38,  
[5] R. P. Shimpi, H. G. Patel and H. Arya (2007), New first-order shear deformation plate theories, J. Appl. Mech., Vol. 74, pp. 523-533  
[6] M. Levinson (1980), An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Commun. Vol. 7, pp. 343-350  
[7] J. N. Reddy (1984), A simple higher-order theory for laminated composite plates, J. Appl. Mech., Vol. 51, pp. 745-752  
[8] R. P. Shimpi (2002), Refined plate theory and its vari-ants, AIAA J, Vol. 40, pp. 137-146  
[9] C.M. Wang, J.N. Reddy, K.H. Lee (2000), Shear De-formable Beams and Plates: Relationships with classical solutions, ISBN 978-0-08-043784-2  
[10] Ajay R.P., S.P. Chandresha, K.B. Parikh (2017), A review paper on analysis and cost-comparison of box culvert for different aspect ratio of cell, International Journal of Engineering Trends and Technology, 44 (3), pp112-115  
[11] M.G. Kalyanshetti, S.V. Malkhare (2012), Analysis of box considering soil structure interaction, Indian Journal of Research, 1(4), pp71-74  
[12] W.D. Lawson, H. Seo, J.G. Surles, S.M. Morse (2017), Culvert Rating Guide, Second Edition, Texas Department of Transportation, pp1-196  
[13] Oladejo J.S., Adetoye O.A. (2022), Reliability evalua-tion of reinforced concrete box culvert at Qua Plateau State Nigeria, International Journal of Research Publica-tion and reviews, 3(6), pp2884-2893  
[14] T. Deshmukh, V. Kadlag (2022), Analysis of box cul-vert under cushion loading, International Journal of Inno-vative Research in science Engineering and Technology (IJIRSET), 11(6), pp7735-7742  
[15] Prema S.B., Guruprasad T.N., T.V. Mallesh, S.R. Ramesh (2019), Parametric study on behaviour of RCC box culvert for dynamic loading, International Research Journal of Engineering and Technology (IRJET), 6(9), pp96-103  
[16] A.D. Patil, A.A. Galatage (2016), Analysis of box cul-vert under cushion loading, International Advanced Re-search Journal in Science Engineering and Technology (IARJSET), 3(6), pp163-166  
[17] Saurav, I. Pandey (2017), Economic design of RCC Box culvert through comparative study of conventional and finite element method, International Journal of Engi-neering and Technology (IJET), 9(3), pp1707-1713  
[18] Filonenko-Borodich MM (1940), Some approximate theories of elastic foundation, Uchenyie Zapiski Mos-kovskogo Gosudarstvennogo Universitet, Mekhanika, 46, 3-18. (in Russian)  
[19] Hetenyi M. (1950), A general solution for the bending of beams on an elastic foundation of arbitrary continuity, Journal of Applied Physics, 21(1), 55-58  
[20] Pasternak PL. (1954), New method of calculation for flexible substructures on two-parameter elastic foundation, Gosudarstvennoe Izdatelstoo. Literatury po Stroitelstvu i Architekture, Moskau, 1-56. (in Russian).  
[21] Reissner E. (1958), A note on deflections of plates on a viscoelastic foundation, Journal of Applied Mechanics ASME, 25(1):144-145  
[22] Mohamed El-Hebib GUELLIL (2012), Modeling of the behaviour of plates resting on elastic soil by the finite element method, Master thesis, Hassiba Ben Bouali Uni-versity of Chlef, pp.1-175.  
[23] Rabinovivi, A. (1970), Reciprocal action between the structure and the foundation soil, Technical bulletin of French-speaking Switzerland, 9, pp. 131-137  
[24] Kerr A.D. (1964), Elastic and Viscoelastic foundation models, Journal of Applied Mechanics, 31, pp491- 498  
[25] Vallabhan CVG, Daloglu AT. (1999), Consistent FEM-Vlasov model for plates on layered soil, Journal of Struc-tural Engineering ASCE, 125(1), pp108-113  
[26] Vlasov VZ, Leont'ev NN (1960), Beams, plates and shells on elastic foundations. GIFML, Moskau. (in Russian)  
[27] G.R. Liu, Y.T. Gu (2005), An Introduction to Meshfree Methods and Their Programming Springer, Dordrecht, The Netherlands  
[28] T. Liszka, J. Orkisz (1980), The finite difference meth-ods at arbitrary irregular grids and its applications in ap-plied mechanics, Computers & Structures, 11, pp. 83-95  
[29] L. Lucy (1977), A numerical approach to testing the fission hypothesis, The Astronomical Journal, 82, pp. 1013-1024  
[30] G.R. Liu, M.B. Liu (2003), Smoothed Particle Hydro-dynamics - A Meshfree Practical Method, World Scien-tific, Singapore, 472page  
[31] R.L. Hardy (1990), Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988), Computers & Mathematics with Applica-tions, 19, pp. 163- 208  
[32] T. Belytschko, Y.Y. Lu, L. Gu (1994), Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 37, pp. 229-256  
[33] S.N. Atluri, T. Zhu (1998), A new meshless local Pe-trov-Galerkin (MLPG) approach in computational me-chanics, Computational Mechanics, 22, pp. 117-127  
[34] G.R. Liu, Y.T. Gu (2001), A point interpolation meth-od for two-dimensional solids, International Journal for Numerical Methods in Engineering, 50, pp. 937-951<937::AID-NME62>3.0.CO;2-X  
[35] G.R. Liu (2002), Meshfree Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, USA  
[36] G.R. Liu, Y.T. Gu (2003), A meshfree method: meshfree weak-strong (MWS) form method, for 2-D sol-ids, Computational Mechanics, 33 (1), pp. 2-14  
[37] G.R. Liu, Y.L. Wu, H. Ding (2004), Meshfree weak-strong (MWS) from method and its application to incom-pressible flow problems, International Journal for Numeri-cal Methods in Fluids, 46, pp. 1025-1047  
[38] L. Cueto-Felgueroso, I. Colominas, F. Navarrina, M. Casteleiro (2005), Numerical simulation of free surface flows by Lagrangian particle methods, Journal of Fluid Structure Interaction and Moving Bounadary Problems, 84, pp. 1-9  
[39] G. Shobeyri, M.H. Afshar (2010), Simulating free surface problems using Discrete Least Squares Meshless method, Computers and Fluids Journal, pp461-470  
[40] F. Daneshmand, M. Kazemzadeh-Parsi (2004), A Meshless Method for Free Surface Flow Through Sluice Gates, 6th International Conference on Hydroinformatics, pp71-78  
[41] W.C. Moon, H.T. Puay, T.L. Lau (2018), Numerical simulation of free surface flow using a multiphase model with higher order scheme, Conference in Advances in civil engineering and science technology, pp 1-8  
[42] V.C. Loukopoulos, G.C. Bourantas, E.D. Skouras (2011), Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions, Computational Mechanics Journal, 47, pp137-159  
[43] Philip F.H., Matthias J.R. (2016), Accurate meshless methods for magnetohydrodynamics, Monthly Notices of the Royal Astronomical Society; 455(1), pp51-88  
[44] B.K. Ferezgdhi, R. Naderi (2017), A Numerical Model-ing for Underwater Explosion Using Mesh-Less Smooth Particle Hydrodynamics Method, Advanced Defence Sci. and Tech., 7; pp161-169  
[45] D. Hu, L. Chunhan, X. YiHua, X. Han (2014), Analy-sis of explosion in concrete by axisymmetric FE-SPH adaptive coupling method; Engineering Computations, 31(4), pp 758-774  
[46] M. Afrasiabi, M. Roethlin, K. Wegener (2019), Con-temporary Meshfree Methods for Three Dimensional Heat Conduction Problems, Archives of Computational Meth-ods in Engineering, 27; pp1413-1447  
[47] R.J. Cheng, K.M. Liew (2012), A meshless analysis of three-dimensional transient heat conduction problems, Engineering Analysis with Boundary Elements, 36(2), pp203-210  
[48] M.T. Mohammadi Anaei, A. Khosravifard, T.Q. Bui (2021), Analysis of fracture mechanics and fatigue crack growth in moderately thick plates using an efficient meshfree approach, Theoretical and Applied Fracture Mechanics, 113; 102943  
[49] N. Fallah, N. Nikraftar (2018), Meshless finite volume method for the analysis of fracture problems in orthotropic media, Engineering Fracture Mechanics, 204, pp46-62  
[50] S.M. Hosseini, C. Zhang (2021), Band structure analy-sis of Green-Naghdi-based thermoelastic wave propagation in cylindrical phononic crystals with energy dissipation using a meshless collocation method, International Jour-nal of Mechanical Sciences, 209; 106711  
[51] J. Ma, H. Gao, G. Wei, J. Qiao (2020), The meshless analysis of wave propagation based on the Hermit-type RRKPM, Soil Dynamics and Earthquake Engineering, 134, 106154  
[52] M.N. Rasoulizadeh, M.J. Ebadi, Z. Avazzadeh, O. Nikan (2021), An efficient local meshless method for the equal width equation in fluid mechanics, Engineering Analysis with Boundary Elements, 131; pp258-268  
[53] S. Shahriari, D. Garcia (2018), Meshfree simulations of ultrasound vector flow imaging using smoothed particle hydrodynamics, Physics in Medicine and Biology, 63(20), pp1-18  
[54] J.F. Wang, J.P. Yang, S.K. Lai, W.Zhang (2020), Sto-chastic meshless method for nonlinear vibration analysis of composite plate reinforced with carbon fibers, Aero-space Science and Technology, 105, 105919  
[55] N. Fallah, M. Delzendeh (2018), Free vibration analy-sis of laminated composite plates using meshless finite volume method, Engineering Analysis with Boundary Elements, pp132-144  
[56] S. Hosseini, G. Rahimi, Y. Anani (2021), A meshless collocation method based on radial basis functions for free and forced vibration analysis of functionally graded plates using FSDT, Engineering Analysis with Boundary Ele-ments, 125, pp168-177  
[57] J. Li, Y. Guan, G. Wang, G. Zhao, J. Lin, H. Naceur, D. Coutellier (2018), Meshless modeling of bending behaviour of bi-directional functionally graded beam structures, Composites Part B: Engineering, 115; pp 104-111  
[58] G. Giunta, S. Belouettar, A.J.M. Ferreira (2016), A static analysis of three-dimensional functionally graded beams by hierarchical modelling and a collocation mesh-less solution method, Acta Mechanica, 227, pp969-991  
[59] H. Mellouli, H. Jrad, M. Wali, F. Dammak (2019), Meshless implementation of arbitrary 3D-shell structures based on a modified first order shear deformation theory, Computers and Mathematics with Applications, 77(1): pp34-39  
[60] K.M. Liew, X. Zhao, A.J.M. Ferreira (2011), A review of meshless methods for laminated and functionally grad-ed plates and shells, Composite Structures, 93(8): pp2031-2041  
[61] P. Jankowski, K.K. Zur, A. Farajpour (2022), Analyti-cal and meshless DQM approaches to free vibration analy-sis of symmetric FGM porous nanobeams with piezoelec-tric effect, Engineering Analysis with Boundary Elements, 136, pp266-289  
[62] K. Kiani (2022), Nonlocal-integro-surface energy-vibro analysis of twist in coaxially composite wire-like nanostructures with internal and interfacial defects via a meshless technique, Engineering Analysis with Boundary Elements, 135, pp217-232  
[63] X. Wang, H. Qi, Z. Sun, L. Hu (2019), A van der Waals contact-bond model for low-dimensional nanoscale car-bon materials based on the quasi-continuum method, Journal of Materials Research, 34(24): pp 4011-4023  
[64] S. Saitta, R. Luciano, R. Vescovini, N. Fantuzzi, F. Fabbrocino (2022), Optimization of a Radial Point Inter-polation Meshless strategy for strain gradient nanoplates, Engineering Analysis with Boundary Elements, 140, pp70-78  
[65] M. Rezaiee-Pajand, M. Mokhtari (2019), A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams, Journal of the Brazilian Society of Mechanical Sciences and Engineer-ing, 41, 303  
[66] R. Ansari, A. Arjangpay (2014), Nanoscale vibration and buckling of single-walled carbon nanotubes using the meshless local Petrov-Galerkin method, Physica E: Low-dimensional Systems and nanostructures, 63, pp283-292  
[67] G. R. Liu (2016), An Overview on Meshfree Methods: For Computational Solid Mechanics, Int J of Computa-tional Methods, 13(5):1-42  
[68] Liu, G. R., Zhang G. Y., Gu Y. T. and Wang Y. Y. (2005), A meshfree radial point interpolation method (RPIM) for three-dimensional solids, Comput Mech, 36, pp421-430.  
[69] Wang, J. G. and Liu, G. R. (2002a), A point interpo-lation meshless method based on radial basis functions, Int. J. Numer. Meth. Engng, 54, pp1623-1648.  
[70] Wang, J. G. and Liu, G. R. (2002b), On the optimal shape parameters of radial basis functions used for 2D meshless methods, Comput. Methods Appl. Mech. Engrg., vol. 191, pp. 2611-2630.  
[71] Gu YT, Liu GR (2001), A local point interpolation method for static and dynamic analysis of thin beams, Comput Methods Appl Mech Eng, 190, pp5515-5528  
[72] Liu GR, Gu YT (2001b), A local radial point interpola-tion method(LR-PIM) for free vibration analyses of 2-D solids. J Sound Vib, 246(1), pp29-46  
[73] Liu GR, Yan L, Wang JG, Gu YT (2002), Point interpo-lation method based on local residual formulation using radial basis functions, Struct Eng Mech, 14(6), pp713-732  
[74] Dinis, L.M.J.S., Natal Jorgea, R.M. and Belinha, J. (2008), Analysis of plates and laminates using the natural neighbour radial point interpolation method, Engineering Analysis with Boundary Elements, 32, 267 -279.  
[75] Liu, Y., Hon Y. C. and Liew K. M. (2006), A meshfree Hermite-type radial point interpolation method for Kirch-hoff plate problems, Int. J. Numer. Meth. Engng, 66: 1153-1178.  
[76] Liu, L., Chua L.P. and Ghista D.N. (2006), Conform-ing radial point interpolation method for spatial shell struc-tures on the stress-resultant shell theory, Arch Appl Mech, 75, 248-267.  
[77] Zhao, X., Liu, G. R., Dai K. Y., Zhong Z. H., Li G. Y. and Han X. (2009a), Free-vibration analysis of shells via a linearly conforming radial point interpolation method (LC-RPIM), Finite Elements in Analysis and Design, 45, pp917-924.  
[78] Zhao, X., Liu, G. R., Dai K. Y., Zhong Z. H., Li G. Y. and Han X. (2009b), A linearly conforming radial point interpolation method (LC-RPIM) for shells, Comput Mech, 43, pp403-413  
[79] Liu L, Liu GR, Tan VBC (2002) Element free method for static and free vibration analysis of spatial thin shell structures. Comput Methods Appl Mech Eng 191(51-52):5923-5942  
[80] Omar A, Said M, Bouazza B (2020) On the use of Radial Point Interpolation Method (RPIM) in a high order continuation for the resolution of the geometrically nonlin-ear elasticity problems. Engineering Analysis with Bound-ary Elements, vol 110, pp 69-79  
[81] Wang JG, Liu GR, Lin P (2002) Numerical analysis of Biot's consolidation process by radial point interpolation method. Int J Solids Struct, 39(6), pp1557-1573  
[82] G. R. Liu, G. Y. Zhang, Y. T. Gu, Y. Y. Wang (2005) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech, 36, pp421-430  
[83] Code Aster (2016), Plate elements: modeling DKT, DST, DKTG and Q4g, Version 12, pp.1-58  
[84] Bui Hung Cuong (2008), Static analysis of the behav-iour of thin-walled structures by the finite element method and finite strips of the plate type and lowered shell de-formable in shear, PhD Thesis, University of Lierge, pp.1-209  
[85] R. Buczkowski, W. Torbacki (2009), Finite element analysis of plate on layered tensionless foundation, Ar-chives of Civil Engineering, LVI, 3, pp. 255-274  
[86] Vallabhan CVG, Straughan W.T, Das YC. (1991), Refined model for analysis of plates on elastic founda-tions, Journal of Engineering Mechanics ASCE, 117(12):2830-2844  
[87] A. Khosravifard and M. R. Hematiyan (2010), A new method for meshless integration in 2D and 3D Galerkin meshfree methods, Engineering Analysis with Boundary Elements, 34, pp. 30-40  
[88] T. Q. Bui and Ch. Zhang (2011), A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis, Computer Methods in Applied Mechanics and Engineering, 200, 1354-1366.  
[89] O. Garcia, E. A. Fancello, C. S. de Barcellos et C. A. Duarte (2000), hp-Clouds in Mindlin's thick plate model, International Journal for Numerical Methods in Engineer-ing, 47(8),1381-1400<1381::AID-NME833>3.0.CO;2-9  
[90] P. de T. R. Mendonc¸a, C. S. de Barcellos, et A. Duarte (2000), In vestigetions on the hp-cloud method by solving timoshenko beam problems, Computational Mechanics, 25(2-3):286-295  
[91] T.Q. Bui and al. (2016), High frequency modes meshfree analysis of Reissner-Mindlin plates, Advanced Materials and Devices, 1, pp. 400-412  
[92] MESSI Alfred (2011), Dimensioning of an ordinary rectangular box culvert, Project, pp. 1-47  
[93] S. Imanzadeh, A. Marache, A. Denis (2011), Estima-tion of the variability of the reaction modulus for the study of the behaviour of strip footings on elastic soil: Applica-tion from existing models, XXIXe Civil Engineering Uni-versity Meetings, Tlemcen, pp. 1-10  
[94] D. Cho, B.H. Kim, J. Kim, N. Vladimir, T. Choi (2017), Simplified dynamic analysis of stepped thickness rectan-gular plate structures by the assumed mode method, Part M: Journal of Engineering for the Maritime Environment, 231(1), pp. 177-187  
[95] Kiani, K., Gharebaghi, S.A. and Mehri, B. (2017), In-plane and out-of-plane waves in nanoplates immersed in bidirectional magnetic fields, Structural engineering and mechanics: An international journal, 61(1), pp.65-76.