A Finite Volume Formulation for the Elasto-Plastic Analysis of Rectangular Mindlin-Reissner Plates, a Non-Layered Approach

Document Type : Research Papers


1 Guilan university

2 University of Guilan


This paper extends the previous work of authors and presents a non-layered Finite Volume formulation for the elasto-plastic analysis of Mindlin-Reissner plates. The incremental algorithm of the elasto-plastic solution procedure is shown in detail. The performance of the formulation is examined by analyzing of plates with different boundary conditions and loading types. The results are illustrated and compared with the predictions of the layered approach. These several comparisons reveal that the non-layered Finite Volume approach can present accurate results with low CPU time usage despite its simplicity of the solution procedure.


Main Subjects

Aguirre, M., Gil, AJ., Bonet, J. and Lee, CH. (2015). “An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics”, Journal of Computational Physics, 300, 387-422.
Bailey, C. and Cross, M. (1995). “A Finite Volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh”, International Journal for Numerical Methods in Engineering, 38(10), 1757-1776.
Basic ́, H., Demirdzˇic ́, I. and Muzaferija, S. (2005). “Finite volume method for simulation of extrusion processes”, International Journal for Numerical Methods in Engineering, 62(4), 475-494.
Bathe, K.J. (1996). Finite Element procedures, Prentice-Hall, New Jersey, USA.
Cardiff, P., Tuković, Ž., Jasak, H. and Ivanković, A. (2016). “A block-coupled Finite Volume methodology for linear elasticity and unstructured meshes”, Computers and Structures, 175, 100-122.
Cardiff, P., Tuković, Ž., De Jaeger, P., Clancy, M. and Ivanković, A. (2017). “A lagrangian cell-centred Finite Volume method for metal forming simulation”, International Journal for Numerical Methods in Engineering, 109(13), 1777-1803.
Cavalcante, M.A.A., Marques, S.P.C. and Pindera, M-J. (2011). “Transient Finite Volume analysis of a graded cylindrical shell under thermal shock loading”, Mechanics of Advanced Materials and Structures, 18(1), 53-67.
Cavalcante, M.A.A., Pindera, M-J. and Khatam, H. (2012). “Finite-Volume micromechanics of periodic materials: Past, present and future”, Composites B, 43(6), 2521-2543.
Cavalcante, M.A.A. and Pindera, M-J. (2012). “Generalized Finite-Volume theory for elastic stress analysis in solid mechanics, Part I: Framework”, Journal of Applied Mechanics, 79(5), 051006-11.
Demirdzˇic ́, I., Martinovic ́, D. and Ivankovic ́, A. (1988). “Numerical simulation of thermal deformation in welded workpiece”, Zavarivanje, 31, 209-219, (in Croatian).
Demirdžić, I. and Martinović, D. (1993). “Finite volume method for thermo-elasto-plastic stress analysis”, Computer Methods in Applied Mechanics and Engineering, 109(3-4), 331-349.
Edalati, H.  and Soltani, B. (2015). “Analysis of thin isotropic and orthotropic plates with Element-Free Galerkin method and various geometric shapes”, Journal of Computational Methods in Engineering, 34(2), 143-157.
Fallah, N. (2004). “A cell vertex and cell centred Finite Volume method for plate bending analysis”, Computer Methods in Applied Mechanics and Engineering, 193(33-35), 3457-3470.
Fallah, N. (2006). “On the use of shape functions in the cell centered Finite Volume formulation for plate bending analysis based on Mindlin-Reissner plate theory”, Computers and Structures, 84(26-27), 1664-1672.
Fallah, N. and Parayandeh-Shahrestany, A. (2014). “A novel Finite Volume based formulation for the elasto-plastic analysis of plates”, Thin-Walled Structure, 77, 153-164.
Ghannadiasl, A.  and Noorzad, A. (2016). “Bending solution for simply supported annular plates using the Indirect Trefftz Boundary method”, Civil Engineering Infrastructures Journal, 49(1), 127-138.
Harrison, D., Ward, T.J.W. and Whiteman, J.R. (1984). “Finite Element analysis of plates with nonlinear properties”, Computer Methods in Applied Mechanics and Engineering, 34(1-3), 1019-1034.
Hughes, T.J.R., Cohen, M. and Haroun, M. (1978). “Reduced and selective integration techniques in the Finite Element analysis of plates”, Nuclear Engineering and design, 46(1), 203-222.
Ivankovic, A., Demirdzic, I., Williams, J.G. and Leevers, P.S. (1994). “Application of the Finite Volume method to the analysis of dynamic fracture problems”, International Journal of Fracture, 66 (4), 357-371.
Kim, S.E., Thai, H.T. and Lee, J. (2009). “Buckling analysis of plates using the two variable refined plate theory”, Thin-Walled Structures, 47(4), 455-462.
Liu, G.R. (2010). Meshfree methods: Moving beyond the Finite Element method second edition, CRC Press, New York, USA. 
Mikaeeli, S.  and Behjat, Ba. (2016). “3-D bending analysis of thick functionally graded plate in different boundary conditions using Element-Free Galerkin (EFG) method”, Journal of Solid and Fluid Mechanics, 6(2), 109-120.
Mirzaei, S., Azhari, M. and Rahim, S.H. (2015). “On the use of finite strip method for buckling analysis of moderately thick plate by refined plate theory and using new types of functions”, Latin American Journal of Solids and Structures, 12(3), 561-582.
Mirzapour, A., Eskandari Ghadi, M. and Ardeshir-Behrestaghi, A. (2012). “Analysis of transversely isotropic half-spaces under the effect of bending of a rigid circular plate”, Civil Engineering Infrastructures Journal, 45(5), 601-610.
Naderi, A. and Baradaran, G.H. (2013). “Element free Galerkin method for static analysis of thin micro/nanoscale plates based on the nonlocal plate theory”, International Journal of Engineering, 26(7), 795-806.
Nordbotten, J.M. (2014). “Cell-centered Finite Volume discretizations for deformable porous media”, International Journal for Numerical Methods in Engineering, 100(9), 399-418.
Osadebe, N. N. and Aginam, C. H. (2011). “Bending analysis of isotropic rectangular plate with all edges clamped: Variational symbolic solution”, Journal of Emerging Trends in Engineering and Applied Sciences, 2(5), 846-852.
Owen, D.R.J. and Hinton, E. (1980). Finite Elements in plasticity: Theory and practice, Pineridge Press, UK.
Prathap, G. and Bhashyam, G.R. (1982). “Reduced integration and the shear-flexible beam element”, International Journal for Numerical Methods in Engineering, 18(2), 195-210.
Reddy, J.N. (2004). Mechanics of laminated composite plates and shells: Theory and analysis, CRC Press, Boca Raton, FL, Second Edition.
Reissner, E. (1945). “The effect of transverse shear deformation on the bending of elastic plates”, ASME Journal of Applied Mechanics, 12, 69-77.
Rezaiee-Pajand, M. and Sadeghi, Y. (2013). “A bending element for isotropic, multilayered and piezoelectric plates”, Latin American Journal of Solids and Structures, 10(2), 323-348.
Ruocco, E. and Fraldi, M. (2012). “An analytical model for the buckling of plates under mixed boundary conditions”, Engineering Structures, 38, 78–88.
Shahabian, F., Elachachi, S.M. and Breysse, D. (2013). “Safety analysis of the patch load resistance of plate girders: Influence of model error and variability”, Civil Engineering Infrastructures Journal, 46(2), 145-160.
Shi, G. and Voyiadjis, G.Z. (1992). “A simple non-layered Finite Element for the elasto-plastic analysis of shear flexible plates”, International Journal for Numerical Methods in Engineering, 33(1), 85-99.
Stylianou, V. and Ivankovic, A. (2002). “Finite Volume analysis of dynamic fracture phenomena”, International Journal of Fracture, 113(2), 107-123.
Sudhir, N. (2012). “Plate bending analysis using Finite Element method”, Bachelor Thesis, Deemed University.
Tang, T., Hededal, O. and Cardiff, P. (2015). “On Finite Volume method implementation of poro-elasto-plasticity soil model”, International Journal for Numerical and Analytical Methods in Geomechanics, 39(13), 1410-1430.
Timoshenko, S.P. and Woinowsky-Krieger, S. (1970). Theory of plates and shells, McGraw-Hill, New York.
Trangenstein, JA. (1991). “The riemann problem for longitudinal motion in an elastic-plastic bar”, SIAM Journal on Scientific and Statistical Computing, 12(1), 180-207.
Wheel, M.A. (1997). “A Finite Volume method for analyzing the bending deformation of thick and thin plates”, Computer Methods in Applied Mechanics and Engineering, 147(1-2), 199-208.
Xia, P., Long, S.Y. and Wei, K.X. (2011). “An analysis for the elasto-plastic problem of the moderately thick plate using the meshless local Petrov Galerkin method”, Engineering Analysis with Boundary Elements, 35(7), 908-914.
Xu, Y. P. and Zhou, D. (2010). “Three-dimensional elasticity solution of transversely isotropic rectangular plates with variable thickness”, Iranian Journal of Science and Technology, 34(B4), 353-369.
Volume 50, Issue 2
December 2017
Pages 293-310
  • Receive Date: 27 November 2016
  • Revise Date: 01 August 2017
  • Accept Date: 11 August 2017
  • First Publish Date: 01 December 2017