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Locking alleviation in the large displacement analysis of beam elements: the strain split method

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Abstract

This paper proposes a new locking alleviation technique for absolute nodal coordinate formulation (ANCF) beam and plate elements based on a strain split approach. The paper also surveys classical finite element (FE) and ANCF locking alleviation techniques discussed in the literature. Because ANCF beam elements, which allow for the cross-sectional stretch fully capture the Poisson effect, Poisson locking is an issue when such beam elements are considered. The two-dimensional fully parameterized ANCF beam element is primarily used in this investigation because such an element can serve as a good surrogate model for three-dimensional ANCF beams and plates as far as membrane, bending and transverse shearing behavior is concerned. In addition to proposing the strain split method (SSM) for ANCF locking alleviation, this work assesses the ANCF element performance in the cases of higher-order interpolation, enhanced assumed strain method, elastic line method, and the enhanced continuum mechanics approach, and demonstrates the design of the enhanced strain interpolation function by using the shape functions of higher-order ANCF elements. Additionally, a new higher-order ANCF two-dimensional beam element is proposed in order to compare its performance with other finite elements that require the use of other locking alleviation techniques proposed and reviewed in the paper. Finally, several numerical examples are shown to demonstrate the effectiveness of the locking alleviation methods applied to ANCF elements. The purpose of this investigation, apart from proposing a new locking alleviation technique, a new higher-order beam element, and comparing several existing locking alleviation techniques, is to show that dealing with locking in fully parameterized ANCF elements is feasible and that several methods exist to effectively improve the ANCF element performance without sacrificing important ANCF element properties and features including position vector gradient continuity. Because of the use of ANCF position vector gradients as nodal coordinates, complex stress-free initially-curved geometries can be systematically obtained. Such initially-curved geometries require special attention when attempting to solve locking problems, as will be discussed in this paper.

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References

  1. Andelfinger, U., Ramm, E.: EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Methods Eng. 36(8), 1311–1337 (1993)

    Article  MATH  Google Scholar 

  2. Armero, F.: On the locking and stability of finite elements in finite deformation plane strain problems. Comput. Struct. 75(3), 261–290 (2000)

    Article  Google Scholar 

  3. Babuska, I., Suri, M.: Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62(1), 439–463 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuska, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29(5), 1261–1293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bathe, K.J.: The inf-sup condition and its evaluation for mixed finite element methods. Comput. Struct. 79(2), 243–252 (2001)

    Article  MathSciNet  Google Scholar 

  6. Bazeley, G.P., Cheung, Y.K., Irons, B.M., Zienkiewicz, O.C. (1965) Triangular elements in plate bending—conforming and nonconforming solutions. In: Proceedings of First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson ATBFB, Ohio

  7. Belytschko, T., Stolarski, H., Liu, W.K., Carpenter, N., Ong, J.S.J.: Stress projection for membrane and shear locking in shell finite elements. Comput. Methods Appl. Mech. Eng. 51(1–3), 221–258 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bergan, P.G., Nygard, M.K.: Finite elements with increased freedom in choosing shape functions. Int. J. Numer. Meth. Eng. 20(4), 643–663 (1984)

    Article  MATH  Google Scholar 

  9. Bischoff, M., Ramm, E.: Shear deformable shell elements for large strains and rotations. Int. J. Numer. Meth. Eng. 40(23), 4427–4449 (1997)

    Article  MATH  Google Scholar 

  10. Bischoff, M., Romero, I.: A generalization of the method of incompatible modes. Int. J. Numer. Methods Eng. 69(9), 1851–1868 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  12. Carpenter, N., Belytschko, T., Stolarski, H.: Locking and shear scaling factors in \(C^{0}\) bending elements. Comput. Struct. 22(1), 39–52 (1986)

    Article  MATH  Google Scholar 

  13. Choi, J., Lim, J.: General curved beam elements based on the assumed strain fields. Comput. Struct. 55(3), 379–386 (1995)

    Article  MATH  Google Scholar 

  14. De Souza Neto, E.A., Peric, D., Owen, D.R.J.: Computational Methods for Plasticity: Theory and Applications. Wiley, West Sussex (2008)

    Book  Google Scholar 

  15. Dmitrochenko, O., Hussein, B.A., Shabana, A.A.: Coupled deformation modes in the large deformation finite element analysis: generalization. ASME J. Comput. Nonlinear Dyn. 4(2), 021002-1–021002-8 (2009)

    Article  Google Scholar 

  16. Dong, S.B., Alpdogan, C., Taciroglu, E.: Much ado about shear correction factors in Timoshenko beam theory. Int. J. Solids Struct. 47(13), 1651–1665 (2010)

    Article  MATH  Google Scholar 

  17. Dorfi, H.R., Busby, H.R.: An effective curved composite beam finite element based on the hybrid-mixed formulation. Comput. Struct. 53(1), 43–52 (1994)

    Article  MATH  Google Scholar 

  18. Dufva, K.E., Sopanen, J.T., Mikkola, A.M.: A two-dimensional shear deformable beam element based on absolute nodal coordinate formulation. J. Sound Vib. 280(3–5), 719–738 (2005)

    Article  Google Scholar 

  19. Ebel, H., Matikainen, M.K., Hurskainen, V., Mikkola, A.: Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity. Nonlinear Dyn. 88(2), 1075–1091 (2016)

    Article  Google Scholar 

  20. Ebel, H., Matikainen, M.K., Hurskainen, V., Mikkola, A.: Higher-order plate elements for large deformation analysis in multibody applications. In: Proceedings of the ASME 2016 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Charlotte, North Carolina, USA, August 21–24 (2016)

  21. Felippa, C.A.: The extended free formulation of finite elements in linear elasticity. ASME J. Appl. Mech. 56(3), 609–616 (1989)

    Article  MATH  Google Scholar 

  22. Friedman, Z., Kosmatka, J.B.: An improved two-node Timoshenko beam finite element. Comput. Struct. 47(3), 473–481 (1993)

    Article  MATH  Google Scholar 

  23. Garcia-Vallejo, D., Mikkola, A., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50(1), 249–264 (2007)

    Article  MATH  Google Scholar 

  24. Gerstmayr, J., Irschik, H.: On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. J. Sound Vib. 318(3), 461–487 (2008)

    Article  Google Scholar 

  25. Gerstmayr, J., Matikainen, M.: Analysis of stress and strain in the absolute nodal coordinate formulation. Mech. Based Des. Struct. Mach. Int. J. 34(4), 409–430 (2006)

    Article  Google Scholar 

  26. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Sys.Dyn. 20, 359–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. ASME J. Comput. Nonlinear Dyn. 8(3), 031016-1–031016-12 (2013)

    Google Scholar 

  28. Hamed, A.M., Jayakumar, P., Letherwood, M.D., Gorsich, D.J., Recuero, A.M., Shabana, A.A.: Ideal compliant joints and integration of computer aided design and analysis. ASME J. Comput. Nonlinear Dyn. 10(2), 021015-1–021015-14 (2015)

    Google Scholar 

  29. Heyliger, P.R., Reddy, J.N.: A higher-order beam finite element for bending and vibration problems. J. Sound Vib. 126(2), 309–326 (1988)

    Article  MATH  Google Scholar 

  30. Hughes, T.J.R., Cohen, M., Haroun, M.: Reduced and selective integration techniques in the finite element analysis of plates. Nucl. Eng. Des. 46(1), 203–222 (1978)

    Article  Google Scholar 

  31. Hughes, T.J.R., Taylor, R., Kanoknukulchai, W.: A simple and efficient finite element for plate bending. Int. J. Numer. Methods Eng. 11(10), 1529–1543 (1977)

    Article  MATH  Google Scholar 

  32. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, New Jersey (1987)

    MATH  Google Scholar 

  33. Hurskainen, V.T., Matikainen, M.K., Wang, J., Mikkola, A.M.: A planar beam finite-element formulation with individually interpolated shear deformation. ASME J. Comput. Nonlinear Dyn. 12(4), 041007-1–041007-8 (2017)

    Google Scholar 

  34. Hussein, B.A., Sugiyama, H., Shabana, A.A.: Coupled deformation modes in the large deformation finite-element analysis: problem definition. ASME J. Comput. Nonlinear Dyn. 2(2), 146–154 (2007)

    Article  Google Scholar 

  35. Ibrahimbegovic, A., Wilson, E.L.: A modified method of incompatible modes. Int. J. Numer. Methods Biomed. Eng. 7(3), 187–194 (1991)

    MATH  Google Scholar 

  36. Kerkkanen, K.S., Sopanen, J.T., Mikkola, A.M.: A linear beam finite element based on the absolute nodal coordinate formulation. ASME J. Mech. Des. 127(4), 621–630 (2005)

    Article  Google Scholar 

  37. Kim, J.G., Kim, Y.Y.: A new higher-order hybrid-mixed curved beam element. Int. J. Numer. Methods Eng. 43(5), 925–940 (1998)

    Article  MATH  Google Scholar 

  38. Kulkarni, S., Pappalardo, C.M., Shabana, A.A.: Pantograph/catenary contact formulations. ASME J. Vib. Acoust. 139(1), 011010-1–011010-12 (2017)

    Google Scholar 

  39. Lee, P., Sin, H.: Locking-free curved beam element based on curvature. Int. J. Numer. Methods Eng. 37(6), 989–1007 (1994)

    Article  MATH  Google Scholar 

  40. Liu, W.K., Belytschko, T., Chen, J.: Nonlinear versions of flexurally superconvergent elements. Comput. Methods Appl. Mech. Eng. 71(3), 241–258 (1988)

    Article  MATH  Google Scholar 

  41. Liu, C., Tian, Q., Hu, H.: Dynamics of a large scale rigid-flexible multibody system composed of composite laminated plates. Multibody Sys.Dyn. 26(3), 283–305 (2011)

    Article  MATH  Google Scholar 

  42. Malkus, D., Hughes, T.J.R.: Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Eng. 15(1), 63–81 (1978)

    Article  MATH  Google Scholar 

  43. Matikainen, M.K., Dmitrochenko, O., Mikkola, A.: Beam elements with trapezoidal cross section deformation modes based on the absolute nodal coordinate formulation. In: International Conference on Numerical Analysis and Applied Mathematics, Rhodes, Greece, September 19–25 (2010)

  44. Mikkola, A.M., Matikainen, M.K.: Development of elastic forces for a large deformation plate element based on the absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 1(2), 103–108 (2006)

    Article  Google Scholar 

  45. Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9(3), 283–309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Mohamed, A.N.A., Liu, J.: The three-dimensional gradient deficient beam element (beam9) using the absolute nodal coordinate formulation. In: Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Buffalo, New York, USA, August 17–20 (2014)

  47. Nachbagauer, K.: State of the Art of ANCF elements regarding geometric description, interpolation strategies, definition of elastic forces, validation and locking phenomenon in comparison with proposed beam finite elements. Arch. Comput. Methods Eng. 21(3), 293–319 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to static and linearized dynamic examples. ASME J. Comput. Nonlinear Dyn. 8(2), 021004-1–021004-7 (2013)

    Google Scholar 

  49. Nachbagauer, K., Gruber, P., Gerstmayr, J.: A 3D shear deformable finite element based on absolute nodal coordinate formulation. Multibody Dyn. Comput. Methods Appl. Sci. 28, 77–96 (2013)

    Article  MATH  Google Scholar 

  50. Nachbagauer, K., Pechstein, A.S., Irschik, H., Gerstmayr, J.: A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 26(3), 245–263 (2011)

    Article  MATH  Google Scholar 

  51. Noor, A., Peters, J.: Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams. Int. J. Numer. Methods Eng. 17(4), 615–631 (1981)

    Article  MATH  Google Scholar 

  52. Ogden, R.W.: Nonlinear Elastic Deformations. Ellis Harwood Ltd., Chichester (1984)

  53. Olshevskiy, A., Dmitrochenko, O., Kim, C.W.: Three-dimensional solid brick element using slopes in the absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 9(2), 021001-1–021001-10 (2014)

    Google Scholar 

  54. Omar, M.A., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243(3), 565–576 (2001)

    Article  Google Scholar 

  55. Orzechowski, G., Fraczek, J.: Nearly incompressible nonlinear material models in the large deformation analysis of beams using ANCF. Nonlinear Dyn. 82(1), 451–464 (2015)

    Article  MathSciNet  Google Scholar 

  56. Orzechowski, G., Shabana, A.A.: Analysis of warping deformation modes using higher-order ANCF beam element. J. Sound Vib. 363, 428–445 (2016)

    Article  Google Scholar 

  57. Pappalardo, C.M., Wallin, M., Shabana, A.A.: A new ANCF/CRBF fully-parameterized plate finite element. ASME J. Comput. Nonlinear Dyn. 12(3), 031008-1–031008-13 (2017)

    Google Scholar 

  58. Pappalardo, C.M., Yu, Z., Zhang, X., Shabana, A.A.: Rational ANCF thin plate finite element. ASME J. Comput. Nonlinear Dyn. 11(5), 051009-1–051009-15 (2016)

    Google Scholar 

  59. Patel, M., Orzechowski, G., Tian, Q., Shabana, A.A.: A new multibody system approach for tire modeling using ANCF finite elements. Proc. Inst. Mech. Eng. Park K J. Multibody Dyn. 230(1), 69–84 (2016)

    Google Scholar 

  60. Pian, T.H.H.: Finite elements based on consistently assumed stresses and displacements. Finite Elem. Anal. Des. 1(2), 131–140 (1985)

    Article  MATH  Google Scholar 

  61. Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress finite elements. Int. J. Numer. Methods Eng. 20(9), 1685–1695 (1984)

    Article  MATH  Google Scholar 

  62. Prathap, G., Babu, R.: Field-consistent strain interpolations for the quadratic shear flexible beam element. Int. J. Numer. Methods Eng. 23(11), 1973–1984 (1986)

    Article  MATH  Google Scholar 

  63. Prathap, G., Babu, R.: An isoparametric quadratic thick curved beam element. Int. J. Numer. Methods Eng. 23(9), 1583–1600 (1984)

    Article  MATH  Google Scholar 

  64. Prathap, G., Bhashyam, G.R.: Reduced integration and the shear-flexible beam element. Int. J. Numer. Methods Eng. 18(2), 195–210 (1982)

    Article  MATH  Google Scholar 

  65. Rakowski, J.: The interpretation of the shear locking in beam elements. Comput. Struct. 37(5), 769–776 (1990)

    Article  Google Scholar 

  66. Rakowski, J.: A critical analysis of quadratic beam finite elements. Int. J. Numer. Methods Eng. 31(5), 949–966 (1991)

    Article  MATH  Google Scholar 

  67. Raveendranath, P., Singh, G., Pradhan, B.: A two-noded locking-free shear flexible curved beam element. Int. J. Numer. Methods Eng. 44(2), 265–280 (1999)

    Article  MATH  Google Scholar 

  68. Reddy, J.N.: On locking-free shear deformable beam finite elements. Comput. Methods Appl. Mech. Eng. 149(1–4), 113–132 (1997)

    Article  MATH  Google Scholar 

  69. Reddy, J.N., Wang, C.M., Lee, K.H.: Relationships between bending solutions of classical and shear deformation beam theories. Int. J. Solids Struct. 34(26), 3373–3384 (1997)

    Article  MATH  Google Scholar 

  70. Sanborn, G.G., Choi, J., Choi, J.H.: Curve-induced distortion of polynomial space curves, flat-mapped extension modeling, and their impact on ANCF thin-plate elements. Multibody Syst. Dyn. 26(2), 191–211 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  71. Sanborn, G.G., Shabana, A.A.: A rational finite element method based on the absolute nodal coordinate formulation. Nonlinear Dyn. 58, 565–572 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  72. Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Long Beach, CA, September 24–28 (2005)

  73. Schwarze, M., Reese, S.: A reduced integration solid-shell finite element based on the EAS and ANS concept—geometrically linear problems. Int. J. Numer. Methods Eng. 80(10), 1322–1355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  74. Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  75. Shabana, A.A.: Computational Continuum Mechanics, 3rd edn. Wiley, Chichester (2018)

    Book  MATH  Google Scholar 

  76. Shabana, A.A.: Dynamics of Multibody Systems, 4th edn. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  77. Shen, Z., Li, P., Liu, C., Hu, G.: A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation. Nonlinear Dyn. 77(3), 1019–1033 (2014)

    Article  Google Scholar 

  78. Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 33(7), 1413–1449 (1992)

    Article  MATH  Google Scholar 

  79. Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29(8), 1595–1638 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  80. Simo, J.C., Armero, F., Taylor, R.L.: Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput. Methods Appl. Mech. Eng. 110(3–4), 359–386 (1993)

    Article  MATH  Google Scholar 

  81. Sopanen, J., Mikkola, A.: Studies on the stiffness properties of the absolute nodal coordinate formulation for three-dimensional beams. In: Proceedings Of Design Engineering Technical Conferences & Computers & Information in Engineering Conference, Chicago, Illinois, USA, September 2–6, 2003 (2003)

  82. Stolarski, H., Belytschko, T.: On the equivalence of mode decomposition and mixed finite elements based on the Hellinger–Reissner principle. Part 1: theory. Comput. Methods Appl. Mech. Eng. 58(3), 249–263 (1986)

    Article  MATH  Google Scholar 

  83. Stolarski, H., Belytschko, T.: On the equivalence of mode decomposition and mixed finite elements based on the Hellinger–Reissner principle. Part 2: applications. Comput. Methods Appl. Mech. Eng. 58(3), 265–284 (1986)

    Article  MATH  Google Scholar 

  84. Stolarski, H., Belytschko, T.: Shear and membrane locking in curved \(C^{0}\) elements. Comput. Methods Appl. Mech. Eng. 41(3), 279–296 (1983)

    Article  MATH  Google Scholar 

  85. Stolarski, H., Chen, Y.: Assumed strain formulation for the four-node quadrilateral with improved in-plane bending behavior. Int. J. Numer. Methods Eng. 38(8), 1287–1305 (1995)

    Article  MATH  Google Scholar 

  86. Sugiyama, H., Gerstmayr, J., Shabana, A.A.: Deformation modes in the finite element absolute nodal coordinate formulation. J Sound Vib 298(4–5), 1129–1149 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  87. Sugiyama, H., Koyama, H., Yamashita, H.: Gradient deficient curved beam element using the absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 5(2), 021001-1–021001-8 (2010)

    Article  Google Scholar 

  88. Sussman, T., Bathe, K.J.: A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput. Struct. 26(1–2), 357–409 (1987)

    Article  MATH  Google Scholar 

  89. Sussman, T., Bathe, K.J.: Spurious modes in geometrically nonlinear small displacement finite elements with incompatible modes. Comput. Struct. 140, 14–22 (2014)

    Article  Google Scholar 

  90. Taylor, R.L., Beresford, P.J., Wilson, E.L.: A non-conforming element for stress analysis. Int. J. Numer. Methods Eng. 10(6), 1211–1219 (1976)

    Article  MATH  Google Scholar 

  91. Taylor, R.L., Filippou, F.C., Saritas, A., Auricchio, F.: A mixed finite element method for beam and frame problems. Comput. Mech. 31(1), 192–203 (2003)

    Article  MATH  Google Scholar 

  92. Tessler, A., Hughes, T.J.R.: An improved treatment of transverse shear in the Mindlin-type four node quadrilateral element. Comput. Methods Appl. Mech. Eng. 39(3), 311–335 (1983)

    Article  MATH  Google Scholar 

  93. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic beams. Philos. Mag. 41, 744–746 (1921)

    Article  Google Scholar 

  94. Valkeapää, A.I., Yamashita, H., Jayakumar, P., Sugiyama, H.: On the use of elastic middle surface approach in the large deformation analysis of moderately thick shell structures using absolute nodal coordinate formulation. Nonlinear Dyn. 80(3), 1133–1146 (2015)

    Article  MathSciNet  Google Scholar 

  95. Von Dombrowski, S.: Modellierung von Balken bei grossen Verformungen für ein kraftreflektierendes Eingabegerät, Diploma thesis, University Stuttgart & DLR (1997)

  96. Wilson, E.L., Taylor, R.L., Doherty, W.P., Ghaboussi, J.: Incompatible displacement models. In: Fenves, J.S., Perrone, N., Robinson, A.R. (eds.) Numerical and Computer Models in Structural Mechanics, pp. 43–57. Academic Press, New York (1973). https://doi.org/10.1016/B978-0-12-253250-4.50008-7

  97. Wriggers, P., Reese, S.: A note on enhanced strain methods for large deformations. Comput. Methods Appl. Mech. Eng. 135(3–4), 201–209 (1996)

    Article  MATH  Google Scholar 

  98. Yakoub, R.Y., Shabana, A.A.: Three-dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. ASME J. Mech. Des. 123(4), 614–621 (2001)

    Article  Google Scholar 

  99. Yamashita, H., Valkeapää, A.I., Jayakumar, P., Sugiyama, H.: Continuum mechanics based bilinear shear deformable shell element using the absolute nodal coordinate formulation. ASME. J. Comput. Nonlinear Dyn. 10(5), 051012-1–051012-9 (2015)

    Google Scholar 

  100. Yunhua, L.: Explanation and elimination of shear locking and membrane locking with field consistence approach. Comput. Methods Appl. Mech. Eng. 162(1–4), 249–269 (1998)

    Article  MATH  Google Scholar 

  101. Zheng, Y., Shabana, A.A.: A two-dimensional shear deformable ANCF consistent rotation-based formulation beam element. Nonlinear Dyn. 87(2), 1031–1043 (2017)

    Article  Google Scholar 

  102. Zienkiewicz, O.C., Owen, D.R.J., Lee, K.N.: Least square-finite element for elasto-static problems. Use of ‘reduced’ integration. Int. J. Numer. Methods Eng. 8(2), 341–358 (1974)

    Article  MATH  Google Scholar 

  103. Zienkiewicz, O.C., Taylor, R., Too, J.M.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Eng. 3(2), 275–290 (1971)

    Article  MATH  Google Scholar 

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  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Mohil Patel & Ahmed A. Shabana

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Patel, M., Shabana, A.A. Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech 229, 2923–2946 (2018). https://doi.org/10.1007/s00707-018-2131-5

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