Shear-flexure Interaction Frame Model on Kerr-type Foundation ‎for Analysis of Non-ductile RC Members on Foundation

Document Type : Research Paper


1 Department of Civil and Environmental Engineering, Prince of Songkla University, Songkhla, 90112, Thailand

2 Civil Engineering Program, School of Engineering, University of Phayao, Phayao, 56000, Thailand

3 School of Engineering and Technology, Walailak University, Nakhorn Si Thammarat, 80160, Thailand

4 Civil Engineering Program, School of Engineering, University of Phayao, Phayao, 56000, Thailand‎

5 Construction and Building Materials Research Center, Department of Civil Engineering, King Mongkut’s University of Technology North Bangkok,‎ Bangkok, 10800, Thailand


In the present day, non-ductile reinforced concrete (RC) members have still appeared in certain parts of the existing old buildings and bridges. These structures always exhibit shear failure when they have been subjected to seismic loading, resulting in a complicated problem for studying or simulating the behaviors of these structures. The soil-structure interaction is a part of these problems that are of interest and motivated in the current study. Therefore, this paper proposes a novel frame model on the Kerr-type foundation with the inclusion of the shear-flexure interaction for the analysis of the non-ductile RC members resting on the foundation. The proposed model is derived from the displacement-based formulation together with the Timoshenko beam theory. The effects of shear-flexure interaction are taken into account in the proposed model through the shear constitutive law. Finally, two numerical simulations are used to assess the capability, accuracy, and efficiency of the proposed model to characterize non-ductile RC members resting on the foundation. Furthermore, these simulations demonstrate the impact of the shear-flexure interaction on the responses of non-ductile RC members on the foundation.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Zhang, J., Makris, N., Seismic response analysis of highway overcrossings including soil–structure interaction, Earthquake Engineering & Structural Dynamics, 31(11), 2002, 1967-1991.
[2] Xie, Y., DesRoches, R., Sensitivity of seismic demands and fragility estimates of a typical California highway bridge to uncertainties in its soil-structure interaction modeling, Engineering Structures, 189, 2019, 605-617.
[3] Katzenbach, R., Leppla, S., Vogler, M., Seip, M., Kurze, S., Soil-structure-interaction of tunnels and superstructures during construction and service time, Procedia Engineering, 57, 2013, 35-44.
[4] Kontogianni, V., Stiros, S.C., Ground loss and static soil–structure interaction during urban tunnel excavation: Evidence from the excavation of the Athens Metro, Infrastructures, 5(8), 2020, 64.
[5] Gharad, A.M., Sonparote, R.S., Influence of soil-structure interaction on the dynamic response of continuous and integral bridge subjected to moving loads, International Journal of Rail Transportation, 8(3), 2020, 285-306.
[6] Östlund, J.L., Andersson, A., Ülker-Kaustell, M., Battini, J.-M., On the influence of shallow soil strata on the dynamic soil–structure interaction of simply supported high-speed railway bridges, International Journal of Rail Transportation, 9(5), 2021, 405-423.
[7] Nguyen, K.T., Reduced-order model for dynamic soil-pipe interaction analysis, Ph.D. Thesis, California institute of technology, Pasadena, California, USA, 2020.
[8] Jena, S.K., Chakraverty, S., Malikan, M., Sedighi, H.M., Implementation of Hermite–Ritz method and Navier's technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity, Journal of Mechanics of Materials and Structures, 15(3), 2020, 405-434.
[9] Koochi, A., Goharimanesh, M., Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: The energy balance method, Reports in Mechanical Engineering, 2(1), 2021, 41-50.
[10] Abouelregal, A.E., Sedighi, H.M., Faghidian, S.A., Shirazi, A.H., Temperature-dependent physical characteristics of the rotating nonlocal nanobeam subject to a varying heat source and a dynamic load, Facta Universitatis, Series: Mechanical Engineering, 19(4), 2021, 633-656.
[11] Gueguen, P., Bard, P.-Y., Soil-structure and soil-structure-soil interaction: Experimental evidence at the Volvi test site, Journal of Earthquake Engineering, 9(5), 2005, 657-693.
[12] Sapountzakis, E., Kampitsis, A., Inelastic analysis of beams on two-parameter tensionless elastoplastic foundation, Engineering Structures, 48, 2013, 389-401.
[13] Limkatanyu, S., Damrongwiriyanupap, N., Kwon, M., Ponbunyanon, P., Force-based derivation of exact stiffness matrix for beams on Winkler-Pasternak Foundation, ZAMM Journal of Applied Mathematics and Mechanics: Zeitschrift für Angewandte Mathematik und Mechanik, 95(2), 2015, 140-155.
[14] Tabatabaiefar, S.H.R., Fatahi, B., Samali, B., Numerical and experimental investigations on seismic response of building frames under influence of soil-structure interaction, Advances in Structural Engineering, 17(1), 2016, 109-130.
[15] Fathi, A., Sadeghi, A., Azadi, M.R.E., Hoveidae, N., Assessing the soil-structure interaction effects by direct method on the out-of-plane behavior of masonry structures (case study: Arge-Tabriz), Bulletin of Earthquake Engineering, 18, 2020, 6429-6443.
[16] Liu, S., Liu, S., Zhang, W., Lu, Z., Experimental study and numerical simulation on dynamic soil‐structure interaction under earthquake excitations, Soil Dynamics and Earthquake Engineering, 138, 2020, 106333.
[17] Yue, F., A refined model for analysis of beams on two-parameter foundations by iterative method, Mathematical Problems in Engineering, 2021, 5562212.
[18] Riaz, M.R., Motoyama, H., Hori, M., Review of soil-structure interaction based on continuum mechanics theory and use of high performance computing, Geosciences, 11(2), 2021, 72.
[19] Schuricht, F., A new mathematical foundation for contact interactions in continuum physics, Archive for Rational Mechanics and Analysis, 184, 2007, 495-551.
[20] Steinmann, P., Geometrical foundations of continuum mechanics, Springer, Berlin, Heidelberg, German, 2015.
[21] Mullapudi, R., Ayoub, A., Nonlinear finite element modeling of beams on two-parameter foundations, Computers and Geotechnics, 37(3), 2010, 334-342.
[22] Limkatanyu, S., Sae-Long, W., Prachasaree, W., Kwon, M., Improved nonlinear displacement-based beam element on a two-parameter foundation, European Journal of Environmental and Civil Engineering, 19(6), 2015, 649-671.
[23] Elishakoff, I., Tonzani, G.M., Zaza, N., Marzani, A., Contrasting three alternative versions of Timoshenko-Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam, ZAMM Journal of Applied Mathematics and Mechanics: Zeitschrift für Angewandte Mathematik und Mechanik, 98(8), 2018, 1334-1368.
[24] Tran, Q.A., Villard, P., Dias, D., Discrete and continuum numerical modeling of soil arching between piles, International Journal of Geomechanics, 19(2), 2019, 04018195.
[25] Yavari, A., Sarkani, S., Reddy, J.N., Generalized solutions of beams with jump discontinuities on elastic foundations, Archive of Applied Mechanics, 71(9), 2001, 625-639.
[26] Limkatanyu, S., Kuntiyawichai, K., Spacone, E., Kwon, M., Nonlinear Winkler-based beam element with improved displacement shape functions, KSCE Journal of Civil Engineering, 17, 2013, 192-201.
[27] Papachristou, K.S., Sophianopoulos, D.S., Buckling of beams on elastic foundation considering discontinuous (unbonded) contact, International Journal of Mechanics and Applications, 3(1), 2013, 4-12.
[28] Boudaa, S., Khalfallah, S., Bilotta, E., Static interaction analysis between beam and layered soil using a two‑parameter elastic foundation, International Journal of Advanced Structural Engineering, 11, 2019, 21-30.
[29] Sae-Long, W., Limkatanyu, S., Hansapinyo, C., Prachasaree, W., Rungamornrat, J., Kwon, M., Nonlinear flexibility-based beam element on Winkler-Pasternak foundation, Geomechanics and Engineering, 24(4), 2021, 371-388.
[30] Alimoradzadeh, M., Salehi, M., Esfarjani, S.M., Nonlinear dynamic response of an axially functionally graded (AFG) beam resting on nonlinear elastic foundation subjected to moving load, Nonlinear Engineering, 8(1), 2019, 250-260.
[31] Winkler, E., Die Lehre von der und Festigkeit, Dominicus, Prague, Czechoslovakia, 1867.
[32] Filonenko-Borodich, M.M., Some approximate theories of the elastic foundation, Uchenyie Zapiski Moskovkogo Gosudarstuennogo Universiteta Mekhanika, Moscow, 46, 1940, 3-18.
[33] Pasternak, P.L., On a new method of analysis of an elastic foundation by means of two foundation constants, Gosuderevstvennae Izdatlesva Literaturi po Stroitelstvu i Arkihitekture, Moscow, USSR (in Russian), 1954.
[34] Hetényi, M., Beams on elastic foundation: Theory with applications in the fields of civil and mechanical engineering, University of Michigan, USA, 1971.
[35] Zhaohua, F., Cook, R.D., Beam elements on two-parameter elastic foundations, Journal of Engineering Mechanics, 109(6), 1983, 1390-1402.
[36] Hetényi, M., A general solution for the bending of beams on an elastic foundation of arbitrary continuity, Journal of Applied Physics, 21(1), 1950, 55-58.
[37] Kerr, A.D., A study of a new foundation model, Acta Mechanica, 1, 1965, 135-147.
[38] Limkatanyu, S., Prachasaree, W., Damrongwiriyanupap, N., Kwon, M., Jung, W., Exact stiffness for beams on Kerr-type foundation: The virtual force approach, Journal of Applied Mathematics, 2013, 626287.
[39] Cai, J.B., Chen, W.Q., Ye, G.R., Ding, H.J., On natural frequencies of a transversely isotropic cylindrical panel on a Kerr foundation, Journal of Sound and Vibration, 232(5), 2000, 997-1004.
[40] Avramidis, I.E., Morfidis, K., Bending of beams on three-parameter elastic foundation, International Journal of Solids and Structures, 43(2), 2006, 357-375.
[41] Zhang, L., Wu, G.T., Wu, J., A Kerr‐type elastic foundation model for the buckling analysis of a beam bonded on an elastic layer, ZAMM Journal of Applied Mathematics and Mechanics: Zeitschrift für Angewandte Mathematik und Mechanik, 99(10), 2019, e201900162.
[42] Wang, J.-J., Chen, F., Li, D., A simple solution of settlement for low reinforced embankments on Kerr foundation, Yantu Lixue/Rock and Soil Mechanics, 40(1), 2019, 250-259.
[43] Jena, S.K., Chakraverty, S., Malikan, M., Application of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation, Engineering with Computers, 37, 2021, 3569-3589.
[44] Lynn, A.C., Moehle, J.P., Mahin, S.A., Holmes, W.T., Seismic evaluation of existing reinforced concrete building columns, Earthquake Spectra, 12(4), 1996, 715-739.
[45] Sezen, H., Whittaker, A.S., Elwood, K.J., Mosalam, K.M., Performance of reinforced concrete buildings during the August 17, 1999 Kocaeli, Turkey earthquake, and seismic design and construction practice in Turkey, Engineering Structures, 25(1), 2003, 103-114.
[46] Mergos, P. E., Beyer, K., Modelling shear–flexure interaction in equivalent frame models of slender reinforced concrete walls, The Structural Design of Tall and Special Buildings, 23(15), 2014, 1171-1189.
[47] Feng, N., Fu, C., Seismic behavior of nonductile RC frame slotted with corrugated steel plate shear walls, Advances in Civil Engineering, 2021, 6653592.
[48] Sae-Long, W., Limkatanyu, S., Damrongwiriyanupap, N., Shear-flexure-interaction frame element inclusion of bond-slip effect for seismic analysis of non-ductile RC columns, Chiang Mai Journal of Science, 41(1), 2022, 14-26.
[49] Nouali, A., Matallah, M., A simplified approach to assess the size effect on the shear-flexure interaction in RC elements, Engineering Structures, 144, 2017, 151-162.
[50] Panedpojaman, P., Design of steel beams and steel beams with openings, 2nd Edition, Songkhla: Faculty of Engineering, Prince of Songkla University (In Thai), 2020.
[51] Elizalde, H., Cárdenas, D., Delgado-Gutierrez, A., Probst, O., In-plane shear-axial strain coupling formulation for shear-‎deformable composite thin-walled beams, Journal of Applied and Computational Mechanics, 7(2), 2021, 450-469.
[52] Vecchio, F.J., Collins, M.P., The modified compression field theory for reinforced concrete elements subjected to shear, ACI Journal, 83(2), 1986, 219-231.
[53] Guedes, J., Pinto, A.V., A numerical model for shear dominated bridge piers, Proceedings of the Second Italy-Japan Workshop on Seismic Design and Retrofit of Bridges, Rome, Italy, 1997.
[54] Ceresa, P., Petrini, L., Pinho, R., Flexure-shear fiber beam-column elements for modeling frame structures under seismic loading — state of the art, Journal of Earthquake Engineering, 11(sup1), 2007, 46-88.
[55] Priestley, M.J.N., Seible, F., Calvi, G.M., Seismic design and retrofit of bridges, John Wiley & Sons, Inc, 1996.
[56] Ricles, J.M., Yang, Y.-S., Priestley, M.J.N., Modeling nonductile R/C columns for seismic analysis of bridges, Journal of Structural Engineering, 124(4), 1998, 415-425.
[57] Sezen, H., Moehle, J.P., Shear strength model for lightly reinforced concrete columns, Journal of Structural Engineering, 130(11), 2004, 1692-1703.
[58] Beyer, K., Dazio, A., Priestley, M.J.N., Shear deformations of slender reinforced concrete walls under seismic loading, ACI Structural Journal, 108(2), 2011, 167-177.
[59] Biskinis, D.E., Roupakias, G.K., Fardis, M.N., Degradation of shear strength of reinforced concrete members with inelastic cyclic displacements, ACI Structural Journal, 101(6), 2004, 773-783.
[60] Marini, A., Spacone, E., Analysis of reinforced concrete elements including shear effects, ACI Structural Journal, 103(5), 2006, 645-655.
[61] Martinelli, L., Modeling shear-flexure interaction in reinforced concrete elements subjected to cyclic lateral loading, ACI Structural Journal, 105(6), 2008, 675-684.
[62] Ceresa, P., Petrini, L., Pinho, R., Sousa, R., A fibre flexure–shear model for seismic analysis of RC-framed structures, Earthquake Engineering & Structural Dynamics, 38(5), 2009, 565-586.
[63] Feng, D.-C., Wu, G., Sun, Z.-Y., Xu, J.-G., A flexure-shear Timoshenko fiber beam element based on softened damage-plasticity model, Engineering Structures, 140, 2017, 483-497.
[64] Feng, D.-C., Xu, J., An efficient fiber beam-column element considering flexure–shear interaction and anchorage bond-slip effect for cyclic analysis of RC structures, Bulletin of Earthquake Engineering, 16, 2018, 5425-5452.
[65] Sae-Long, W., Limkatanyu, S., Prachasaree, W., Horpibulsuk, S., Panedpojaman, P., Nonlinear frame element with shear–flexure interaction for seismic analysis of non-ductile reinforced concrete columns, International Journal of Concrete Structures and Materials, 13, 2019, 32.
[66] Sae-Long, W., Limkatanyu, S., Hansapinyo, C., Imjai, T., Kwon, M., Forced-based shear-flexure-interaction frame element for nonlinear analysis of non-ductile reinforced concrete columns, Journal of Applied and Computational Mechanics, 6, 2020, 1151-1167.
[67] Mergos, P.E., Kappos, A.J., A distributed shear and flexural flexibility model with shear-flexure interaction for R/C members subjected to seismic loading, Earthquake Engineering & Structural Dynamics, 37(12), 2008, 1349-1370.
[68] Mergos, P.E., Kappos, A.J., A gradual spread inelasticity model for R/C beam-columns, accounting for flexure, shear and anchorage slip, Engineering Structures, 44, 2012, 94-106.
[69] Sae-Long, W., Limkatanyu, S., Panedpojaman, P., Prachasaree, W., Damrongwiriyanupap, N., Kwon, M., Hansapinyo, C., Nonlinear Winkler-based frame element with inclusion of shear-flexure interaction effect for analysis of non-ductile RC members on foundation, Journal of Applied and Computational Mechanics, 7(1), 2021, 148-164.
[70] Carrera, E., Giunta, G., Petrolo, M., Beam structures: Classical and advanced theories, John Wiley & Sons, Ltd., 2011.
[71] Taylor, R.L., FEAP: A finite element analysis program, User manual: version 7.3, Department of Civil and Environmental Engineering, University of California, Berkeley, USA, 2000.
[72] Kent, D.C., Park, R., Flexural members with confined concrete, Journal of the Structural Division, 97(7), 1971, 1964-1990.
[73] Menegotto, M., Pinto, P.E., Method of analysis for cyclically loaded reinforced concrete plane frames including changes in geometry and nonelastic behavior of elements under combined normal force and bending, Proceeding of IABSE Symposium on Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads, Lisbon, 1973, 15-22.
[74] Spacone, E., Limkatanyu, S., Responses of reinforced concrete members including bond-slip effects, ACI Structural Journal, 97(6), 2000, 831-839.
[75] Rakowski, J., The interpretation of the shear locking in beam elements, Computers & Structures, 37(5), 1990, 769-776.
[76] Beirao da Veiga, L., Lovadina, C., Reali, A., Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Computer Methods in Applied Mechanics and Engineering, 241-244, 2012, 38-51.
[77] Lee, S.J., Park, K., Static analysis of Timoshenko beams using isogeometric approach, Architectural Research, 16(2), 2014, 57-65.
[78] Baier-Saip, J.A., Baier, P.A., de Faria, A.R., Oliveira, J.C., Baier, H., Shear locking in one-dimensional finite element methods, European Journal of Mechanics / A Solids, 79, 2020, 103871.
[79] Priestley, M.J.N., Seible, F., Verma, R., Xiao, Y., Seismic shear strength of reinforced concrete columns, Structural Systems Research Project Report No. SSRP 93/06, University of California, San Diego, USA, 1993.
[80] Park, R., Paulay, T., Reinforced concrete structures, John Wiley & Sons, New York, USA, 1975.
[81] Sezen, H., Seismic behavior and modeling of reinforced concrete building columns, Ph.D. Thesis, Department of Civil and Environmental Engineering, University of California, Berkeley, USA, 2002.
[82] Lynn, A.C., Seismic evaluation of existing reinforced concrete building columns, Ph.D. Thesis, Department of Civil and Environmental Engineering, University of California, Berkeley, USA, 2001.
[83] Antonio, L.M., Pavanello, R., de Almeida Barros, P.L., Marine pipeline–seabed interaction modeling based on Kerr-type foundation, Applied Ocean Research, 80, 2018, 228-239.
[84] Shiri, H., Response of steel catenary risers on hysteretic non-linear seabed, Applied Ocean Research, 44, 2014, 20-28.
[85] Alemdar, B.N., Gülkan, P., Beams on generalized foundations: supplementary element matrices, Engineering Structures, 19(11), 1997, 910-920.
[86] Morfidis, K., Exact matrices for beams on three-parameter elastic foundation, Computers & Structures, 85(15-16), 2007, 1243-1256.