Advanced Steel Construction

Vol. 13, No. 3, pp. 206-240 (2017)


A 3-NODE CO-ROTATIONAL TRIANGULAR ELASTO-PLASTIC

SHELL ELEMENT USING VECTORIAL ROTATIONAL

VARIABLES

 

Zhongxue Li1*, Bassam A. Izzuddin2, Loc Vu-Quoc3, Zihan Rong4 and Xin Zhuo4

1,* Department of Civil Engineering, Zhejiang University, China

2Department of Civil and Environmental Engineering, Imperial College London, United Kingdom

3Department of Mechanical and Aerospace Engineering, University of Florida, USA

4Department of Civil Engineering, Zhejiang University, PR China

*(Corresponding author: E-mail:This email address is being protected from spambots. You need JavaScript enabled to view it.)

Received: 21 March 2016; Revised: 31 July 2016; Accepted: 3 August 2016

 

DOI:10.18057/IJASC.2017.13.3.2

 

View Article   Export Citation: Plain Text | RIS | Endnote

ABSTRACT

A 3-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element employs a zero-‘macro spin’ framework at the macro element level, thus reducing the material spin over the element domain, and resulting in an invariance of the element tangent stiffness matrix to the order of the node numbering. The two smallest components of each nodal orientation vector are defined as rotational variables, achieving the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite-element formulations, both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell-Huber-Hencky-von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is used. Assumed membrane strains and assumed shear strains---calculated respectively from the edge-member membrane strains and the edge-member transverse shear strains---are employed to overcome locking problems, and the residual bending flexibility is added to the transverse shear flexibility to improve further the accuracy of the element. The reliability and convergence of the proposed 3-node triangular shell element formulation are verified through two elastic plate patch tests as well as three elastic and three elasto-plastic plate/shell problems undergoing large displacements and large rotations.

 

KEYWORDS

Co-rotational approach, elasto-plasticity, triangular shell element, assumed strain, vectorial rotational variable, zero-‘macro spin’


REFERENCES

[1]        Jeon, H.M., Lee, Y.Y., Lee, P.S. and Bathe, K.J., “The MITC3+ Shell Element in Geometric Nonlinear Analysis”, Computers and Structures, 2015, Vol. 146, pp. 91-  104.

[2]        Lee, Y.Y., Lee, P.S., and Bathe, K.J., “The MITC3+ Shell Element and its Performance, Computers and Structures, 2014, Vol. 138, pp. 12-23.

[3]        Jeon, H.M., Lee, P.S. and Bathe, K.J., “The MITC3 Shell Finite Element Enriched by Interpolation Covers”, Computers and Structures, 2014, Vol. 134, pp. 128-142.

[4]        Lee, P. S. and Bathe, K. J., “Development of MITC Isotropic Triangular Shell Finite Elements”, Computers and Structures, 2004, Vol. 82, No. 11-12, pp. 945–962.

[5]        Lee, P.S. and Bathe, K. J., “The Quadratic MITC Plate and MITC Shell Elements in Plate Bending”, Advances in Engineering Software, 2010, Vol. 41, No. 5, pp. 712–728.

[6]        Laulusa, A., Bauchau, O.A., Choi, J.Y., Tan, V.B.C., and Li, L., “Evaluation of Some Shear Deformable Shell Elements”, International Journal of Solids and Structures, 2006, Vol. 43, No. 17, pp. 5033-5054.

[7]        Chapelle, D. and Suarez, I.P., “Detailed Reliability Assessment of Triangular MITC Elements for Thin Shells”, Computers and Structures, 2008, Vol. 86,, No. 23-24, pp. 2192–2202.

[8]        Lee, P.S., Noh, H.C. and Bathe, K.J., “Insight into 3-node Triangular Shell Finite Elements: The Effects of Element Isotropy and Mesh Patterns”, Computers and Structures, 2007, Vol. 85, No. 7-8, pp. 404–418.

[9]        Mohan, P. and Kapania, R.K., “Updated Lagrangian Formulation of a Flat Triangular Element for Thin Laminated Shells”, AIAA Journal, 1998, Vol. 36, No. 2, pp. 273-281.

[10]      Providasa, E. and Kattis, M.A., “An Assessment of Two Fundamental Flat Triangular Shell Elements with drilling Rotations”, Computers and Structures, 2000, Vol. 77, No. 2, pp. 129-139.

[11]      Khosravi, P., Ganesan, R. and Sedaghati, R., “Corotational Non-linear Analysis of Thin Plates and Shells using a New Shell Element”, International Journal for Numerical Methods in Engineering, 2007, Vol. 69, No. 4, pp. 859–885.

[12]      Khosravi, P., Ganesan, R. and Sedaghati, R., “An Efficient Facet Shell Element for Corotational Nonlinear Analysis of Thin and Moderately Thick Laminated Composite Structures”, Computers and Structures, 2008, Vol. 86, No. 9, pp. 50-858.

[13]      Carpenter, N., Stolarski, H. and Belytschko, T., “Improvements in 3-Node Triangular Shell Elements”, International Journal for Numerical Methods in Engineering, 1986, Vol. 23, No. 9, pp. 1643-1667.

[14]      Sabourin, F. and Brunet, M., “Analysis of Plates and Shells with a Simplified 3-node Triangular Element”, Thin-Walled Structures, 1995, Vol. 21, No. 3, pp. 209-223.

[15]      Sabourin, F., and Brunet, M., “Detailed Formulation of the Rotation-free Triangular Element "S3" for General Purpose Shell Analysis”, Engineering Computations, 2006, Vol. 23, No. 5-6, pp. 469-502.

[16]      Oñate, E. and Zárate, F., “Rotation-free Triangular Plate and Shell Elements”, International Journal for Numerical Methods in Engineering, 2000, Vol. 47, No. 1-3, pp. 557-603.

[17]      Flores, F.G. and Oñate, E., “A Basic Thin Shell Triangle with Only Translational DOFs Large Strain Plasticity”, International Journal for Numerical Methods in Engineering, 2001, Vol. 51, No. 1, pp. 57-83.

[18]      Oñate, E. and Flores, F.G., “Advances in the Formulation of the Rotation-free Basic Shell Triangle”, Computer Methods in Applied Mechanics and Engineering, 2005, Vol. 194, No. 21-24, pp. 2406–2443.

[19]      Flores, F.G. and Oñate, E., “Improvements in the Membrane Behaviour of the Three Node Rotation-free BST Shell Triangle using an Assumed Strain Approach”, Computer Methods in Applied Mechanics and Engineering, 2005, Vol. 194, No. 6-8, pp. 907–932.

[20]      Oñate, E. and Zárate, F., “Extended Rotation-free Plate and Beam Elements with Shear Deformation Effects”, International Journal for Numerical Methods in Engineering, 2010, Vol. 83, No. 2, pp. 196-227.

[21]      Zárate, F. and Oñate., E., “Extended Rotation-free Shell Triangles with Transverse Shear Deformation Effects”, Computational Mechanics, 2012, Vol. 49, No. 4, pp. 487-503.

[22]      Stolarski, H., Gilmanov, A. and Sotiropoulos, F., “Nonlinear Rotation-free Three-node Shell Finite Element Formulation”, International Journal for Numerical Methods in Engineering2013, Vol. 95, No. 9, pp. 740-770.

[23]      Kabir, H. R. H., “A Shear-locking Free Robust Isoparametric 3-node Triangular Element for General Shells”, Computers and Structures, 1994, Vol. 51, No. 4, pp. 425-436.

[24]      Wu, S., Li, G.Y. and Belytschko, T., “A DKT Shell Element for Dynamic Large Deformation Analysis”, Communications in Numerical Methods in Engineering, 2005, Vol. 21, No. 11, pp. 651-674.

[25]      Kuznetsov, V.V. and Levyakov, S.V., “Geometrically Nonlinear Shell Finite Element based on the Geometry of a Planar Curve”, Finite Elements in Analysis and Design, 2008, Vol. 44, No. 8, pp. 450-461.

[26]      Kolahi, A.S. and Crisfield, M.A., “A Large-strain Elasto–plastic Shell Formulation using the Morley Triangle”, International Journal for Numerical Methods in Engineering, 2001, Vol. 52, No. 8, pp. 829–849.

[27]      Providas, E. and Kattis, M.A., A” Simple Finite Element Model for the Geometrically Nonlinear Analysis of Thin Shells”, Computational Mechanics, 1999, Vol. 24, No. 2, pp. 127-137.

[28]      Carpenter, N. and Stolarski, H. and Belytschko, T., “A Flat Triangular Shell Element with Improved Membrane Interpolation”, Communications in Applied Numerical Methods, 1985, Vol. 1, No. 4, pp. 161-168.

[29]      Fish, J. and Belytschko, T., “Stabilized Rapidly Convergent 18-degrees-of-freedom Flat Shell Triangular Element”, International Journal for Numerical Methods in Engineering, 1992, Vol. 33, No. 1, pp. 149-162.

[30]      Allman, D.J., “On the assumed displacement-fields of a shallow curved shell finite-element”, Communications in Numerical Methods in Engineering, 1995, Vol. 11, No. 2, pp. 159-166.

[31]      To, C. W. S. and Wang, B., “Hybrid Strain-based Three-node Flat Triangular Laminated Composite Shell Elements”, Finite Elements in Analysis and Design, 1998, Vol. 28, No. 3, pp. 177-207.

[32]      Fajman, P., “New Triangular Plane Element with Drilling Degrees of Freedom”, Journal of Engineering Mechanics-ASCE, 2002, Vol. 128, No. 4, pp. 413-418.

[33]      Cai, Y.C. and Atluri, S.N., “Large Rotation Analyses of Plate/Shell Structures based on the Primal Variational Principle and a Fully Nonlinear Theory in the Updated Lagrangian Co-rotational Reference Frame”, CMES-Computer Modeling in Engineering & Sciences, 2012, Vol. 83, No. 3, pp. 249-273.

[34]      Zhang, Y.X. and Cheung, Y.K., “A Refined Non-linear Non-conforming Triangular Plate/Shell Element”, International Journal for Numerical Methods in Engineering, 2003, Vol. 56, No. 15, pp. 2387–2408.

[35]      Poulsen, P. and Damkilde, L., “A Flat Triangular Shell Element with Loof Nodes”, International Journal for Numerical Methods in Engineering, 1996, Vol. 39, No. 22, pp.3867-3887.

[36]      Campello, E. M. B., Pimenta, P. M. and Wriggers, P., “A Triangular Finite Shell Element based on a Fully Nonlinear Shell Formulation”, Computational Mechanics, 2003, Vol. 31, No. 6, pp. 505-518.

[37]      Yang, H. T. Y., Saigal, S., Masud, A. and Kapania, R. K., “Survey of Recent Shell Finite Elements, International Journal for Numerical Methods in Engineering, 2000, Vol. 47, No. 1-3, pp. 101-127.

[38]      Reissner, E., “A Note on Variational Theorems in Elasticity”, International Journal of Solids and Structures, 1965, Vol. 1, No. 1, pp. 93-95.

[39]      Hughes, T. J. R. and Brezzi, F., “On Drilling Degrees of Freedom”, Computer Methods in Applied Mechanics and Engineering, 1989, Vol. 72, No. 1, pp. 105-121.

[40]      Hughes, T. J. R., Masud, A. and Harari, I., “Numerical Assessment of Some Membrane Elements with Drilling Degrees of Freedom”, Computers and Structures, 1995, Vol. 55, No. 2, pp. 297-314.

[41]      Hughes, T. J. R., Masud, A. and Harari, I., “Dynamic Analysis with Drilling Degrees of Freedom”, International Journal for Numerical Methods in Engineering, 1995, Vol. 38, No. 19, pp. 3193-3210.

[42]      MacNeal, R.H., “Derivation of Element Stiffness Matrices by Assumed Strain Distribution”, Nuclear Engineering and Design, 1982, Vol. 70, No. 1, pp. 3–12.

[43]      Liu, M. L. and To, C. W. S., “Hybrid Strain-based 3-node Flat Triangular Shell Elements”, 1. Nonlinear-theory and Incremental Formulation”, Computers and Structures, 1995, Vol. 54, No. 6, pp. 1031-1056.

[44]      Liu, M. L. and To, C. W. S., “A Further Study of Hybrid Strain-based Three-node Flat Triangular Shell Elements”, Finite Elements in Analysis and Design, 1998, Vol. 31, No. 2, pp. 135-152.

[45]      To, C. W. S. and Wang, B., “Hybrid Strain-based Three-node Flat Triangular Laminated Composite Shell Elements”, Finite Elements in Analysis and Design, 1998, Vol. 28, No. 3, pp. 177-207.

[46]      Ibrahimbegovic, A. and Wilson, E.L., “A Modified Method of Incompatible Modes”, Communications in Applied Numerical Methods, 1991, Vol. 7, No. 3, pp. 187-194.

[47]      Bischoff, M. and Romero, I., “A Generalization of the Method of Incompatible Modes”, International Journal for Numerical Methods in Engineering, 2007, Vol. 69, No. 9, pp. 1851-1868.

[48]      Liu, W.K., Belytschko, T., Ong, J. S. J. and Law, S. E., “Use of Stabilization Matrices in Non-linear Analysis”, Engineering Computations, 1985, Vol. 2, No. 1, pp. 47-55.

[49]      Belytschko, T., Wong, B.L. and Stolarski, H., “Assumed Strain Stabilization Procedure for the 9-node Lagrange Shell Element”, International Journal for Numerical Methods in Engineering, 1989, Vol. 28, No. 2, pp. 385-414.

[50]      Liu, W. K., Ong, J. S. J. and Uras, R. A., “Finite-element Stabilization Matrices - A Unification Approach”, Computer Methods in Applied Mechanics and Engineering, 1985, Vol. 53, No. 1, pp. 13-46.

[51]      Belytschko, T., Tsay, C.S. and Liu, W.K., “A Stabilization Matrix for the Bilinear Mindlin Plate Element”, Computer Methods in Applied Mechanics and Engineering, 1981, Vol. 29, No. 3, pp. 313-327.

[52]      Masud, A., Tham, C.L., and Liu, W.K., “A Stabilized 3-D Co-rotational Formulation for Geometrically Nonlinear Analysis of Multi-layered Composite Shells”, Computational Mechanics 2000, Vol. 26, No. 1, pp. 1-12.

[53]      Belytschko, T. and Leviathan, I., “Physical Stabilization of the 4-node Shell Element with One-point Quadrature”, Computer Methods in Applied Mechanics and Engineering, 1994, Vol. 113, No. 3-4, pp. 321-350.

[54]      Nguyen-Xuan, H., Liu, G. R. and Thai-Hoang, C. and Nguyen-Thoi, T., “An Edge-based Smoothed Finite Element Method (ES-FEM) with Stabilized Discrete Shear Gap Technique for Analysis of Reissner-Mindlin Plates”, Computer Methods in Applied Mechanics and Engineering, 2010, Vol. 199, No. 9-12, pp. 471-489.

[55]      Argyris, J. H., Papadrakakis, M., Apostolopoulou, C. and Koutsourelakis, S., “The TRIC Shell Element: Theoretical and Numerical Investigation”, Computer Methods in Applied Mechanics and Engineering 2000, Vol. 182, No. 1-2, pp. 217-245.

[56]      Ibrahimbegovic, A., “Finite Elastoplastic Deformations of Space-Curved Membranes”, Computer Methods in Applied Mechanics and Engineering, 1994, Vol. 119, No. 3-4, pp. 371-394.

[57]      El-Metwally, S. E., El-Shahhat, A. M. and Chen, W. F., “3-D Nonlinear-analysis of R/C Slender Columns”, Computers and Structures, 1990, Vol. 37, No. 5, pp. 863-872.

[58]      Izzuddin, B. A. and Elnashai, A.S., “Adaptive Space Frame Analysis .2. AD Distributed Plasticity Approach”, Proceedings of the Institution of Civil Engineers-Structures and Buildings, 1993, Vol. 99, No. 3, pp. 317-326.

[59]      Spacone, E., Filippou, F.C. and Taucer, F.F., “Fibre Beam-column Model for Non-linear Analysis of R/C Frames”, 1. Formulation, Earthquake Engineering & Structural Dynamics, 1996, Vol. 25, No. 7, pp. 711-725.

[60]      Izzuddin, B.A. and Lloyd Smith, D., “Large-displacement Analysis of Elastoplastic Thin-walled Frames” 1. Formulation and Implementation, Journal of Structural Engineering-ASCE, 1996, Vol. 122, No. 8, pp. 905-914.

[61]      Crisfield, M. A., “Nonlinear Finite Element Analysis of Solid and Structures”, Vol 1: Essentials. Wiley: Chichester, 1991.

[62]      Becker, M. and Hackenberg, H. P., “A Constitutive Model for Rate Dependent and Rate Independent inelasticity, Application to IN718, International Journal of Plasticity, 2011, Vol. 27, No. 4, pp. 596-619.

[63]      Simo, J. C. and Hughes, T. J. R., “Computational Inelasticity. Springer: New York, 1998.

[64]      Khan, A. S. and Huang, S. J., “Continuum Theory of Plasticity. John Wiley & Sons, Inc: New York. 1995; P94.

[65]      Izzuddin, B.A. and Liang, Y., “Bisector and Zero-macrospin Co-rotational Systems for Shell Elements”, International Journal for Numerical Methods in Engineering, 2015, Vol. 105, No. 4, pp. 286-320.

[66]      Nour-Omid, B. and Rankin, C.C., “Finite Rotation Analysis and Consistent Linearization using Projectors”, Computer Methods in Applied Mechanics and Engineering, 1991, Vol. 93, No. 3, pp. 353-384.

[67]      Simo, J. C. and Tarnow, N. “A New Energy and Momentum Conserving Algorithm for the Non-linear Dynamics of Shells”, International Journal for Numerical Methods in Engineering, 1994, Vol. 37, No. 15, pp. 2527-2549.

[68]      Simo, J.C., Rifai, M.S. and Fox, D.D., “On a Stress Resultant Geometrically Exact Shell-model .6. Conserving Algorithms for Nonlinear Dynamics”, International Journal for Numerical Methods in Engineering, 1992, Vol. 34, No. 1, pp. 117-164.

[69]      Tan, X. G. and Vu-Quoc, L., “Efficient and Accurate Multilayer Solid-shell Element: Nonlinear Materials at Finite Strain”, International Journal for Numerical Methods in Engineering 2005, Vol. 63, No. 15, pp. 2124-2170.

[70]      Vu-Quoc, L. and Tan, X. G., “Optimal Solid Shells for Nonlinear Analyses of Multilayer Composites”, Part II: Dynamics, Computer Methods in Applied Mechanics and Engineering, 2003, Vol. 192, No. 9-10, pp. 1017-1059.

[71]      Li, Z. X., Zhuo, X., Vu-Quoc, L., Izzuddin, B.A. and Wei, H.Y., “A 4-node Co-rotational Quadrilateral Elasto-plastic Shell Element using Vectorial Rotational Variables”, International Journal for Numerical Methods in Engineering, 2013, Vol. 95, No. 3, pp. 181-211.

[72]      Li, Z. X., Xiang, Y., Izzuddin, B. A., Vu-Quoc, L., Zhuo, X. and Zhang, C.J., “A 6-node Co-rotational Triangular Elasto-plastic Shell Element", Computational Mechanics, 2015, Vol. 55, No. 5, pp. 837-859.

[73]      MacNeal, R. H. and Harder, R. L., “Proposed Standard Set of Problems to Test Finite Element Accuracy”, Finite Elements in Analysis and Design, 1985, Vol. 1, No. 1, pp. 3-20.

[74]      Buechter, N. and Ramm, E., “Shell Theory Versus Degeneration - A Comparison in Large Rotation Finite-element Analysis”, International Journal for Numerical Methods in Engineering, 1992, Vol. 34, No. 1, pp. 39-59.

[75]      Simo, J. C., Fox, D.D. and Rifai, M. S., “On a Stress Resultant Geometrically Exact Shell Model”, Part III: Computational Aspects of the Nonlinear Theory”, Computer Methods in Applied Mechanics and Engineering, 1990, Vol. 79, No. 1, pp. 21–70.

[76]      Jiang, L. and Chernuka, M.W., “A Simple 4-noded Corotational Shell Element for Arbitrarily Large Rotations”, Computers and Structures, 1994, Vol. 53, No. 5, pp. 1123–1132.

[77]      Sze, K.Y., Chan, W.K. and Pian, T. H. H., “An 8-node Hybrid-stress Solid-shell Element for Geometric Nonlinear Analysis of Elastic Shells”, International Journal for Numerical Methods in Engineering, 2000, Vol. 55, No. 7, pp. 853–878.

[78]      Eberlein, R. and Wriggers, P., “Finite Element Concepts for Finite Elastoplastic Strains and Isotropic Stress Response in Shells: Theoretical and Computational Analysis”, Computer Methods in Applied Mechanics and Engineering, 1999, Vol. 171, No. 3-4, pp. 243-279.

[79]      Betsch, P. and Stein, E., “Numerical Implementation of Multiplicative Elasto-plasticity into Assumed Strain Elements with Application to Shells at Large Strains”, Computer Methods in Applied Mechanics and Engineering, 1999, Vol. 179, No. 3-4, pp. 215-245.

[80]      Valente, R. A. F., Parente, M. P. L., Jorge, R. M. N., de Sa, J. M. A. C. and Gracio, J. J., “Enhanced Transverse Shear Strain Shell Formulation Applied to Large Elasto-plastic Deformation Problems”, International Journal for Numerical Methods in Engineering, 2005, Vol. 62, No. 10, pp.1360-1398.

[81]      Miehe, C., “A Theoretical and Computational Model for Isotropic Elastoplastic Stress Analysis in Shells at Large Strains”, Computer Methods in Applied Mechanics and Engineering, 1998, Vol. 155, No. 3-4, pp. 193-233.

[82]      Izzuddin, B. A., “An Enhanced Co-rotational Approach for Large Displacement Analysis of Plates”, International Journal for Numerical Methods in Engineering, 2005, Vol. 64, No. 10, pp. 1350-1374.

[83]      Li, Z. X., “A Co-rotational Formulation for 3D Beam Element using Vectorial Rotational Variables”, Computational Mechanics, 2007, Vol. 39, No. 3, pp. 309-322

[84]      Li, Z. X., and Vu-Quoc, L., “A Mixed Co-rotational 3D Beam Element Formulation for Arbitrarily Large Rotations”, Advanced Steel Construction, 2010, Vol. 6, No. 2, pp. 767-787.

[85]      Li, Z. X., “Bionics and Computational Theory using Advanced Finite Element Methods”, Science Press: Beijing, 2009.

[86]      Tessler, A., and Hughes, T. J. R., “A 3-node Mindlin Plate Element with Improved Transverse-shear”, Computer Methods in Applied Mechanics and Engineering, 1985, Vol. 50, No. 1, pp. 71-101.

[87]      MacNeal, R. H., “Finite Elements: Their Design and Performance”, Marcel Dekker Inc. New York, 1993, pp. 369-417.