Locking-free curved elements with refined kinematics for the analysis of composite structures

Abstract A new class of refined curved beam elements is proposed for the accurate stress analysis of composite structures. The element possesses three-dimensional capabilities and it is suited for the study of curved laminates and fiber-reinforced composites at the microscopic scale. The numerical issues associated with membrane and shear lockings are overcome by means of assumed interpolations of the strain components based on the mixed interpolation of tensorial components method (MITC). Higher-order expansions with only displacement unknowns are employed for the cross-section assumptions at the component level, enabling the computation of component-wise stress fields. For this purpose, a hierarchical set of Legendre functions is implemented, which allows the user to tune the kinematics of the element through the polynomial order input. The detrimental effects of locking in composite modeling are investigated and the robustness and efficiency of the beam element is assessed through comparison against solutions from the literature and refined solid models.

[1]  E. Carrera,et al.  Component-wise analysis of laminated structures by hierarchical refined models with mapping features and enhanced accuracy at layer to fiber-matrix scales , 2018 .

[2]  Yoon Young Kim,et al.  A new higher‐order hybrid‐mixed curved beam element , 1998 .

[3]  K. Bathe.  Finite Element Procedures , 1995 .

[4]  D. Griffin,et al.  Finite-Element Analysis , 1975 .

[5]  Alexander Tessler,et al.  Curved beam elements with penalty relaxation , 1986 .

[6]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[7]  P. Pinsky,et al.  An assumed covariant strain based 9‐node shell element , 1987 .

[8]  E. Carrera,et al.  Extension of MITC to higher‐order beam models and shear locking analysis for compact, thin‐walled, and composite structures , 2017 .

[9]  E. Hinton,et al.  A new nine node degenerated shell element with enhanced membrane and shear interpolation , 1986 .

[10]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[11]  I. Fried,et al.  Shape functions and the accuracy of arch finite elements. , 1973 .

[12]  A. Arpaci,et al.  Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations , 2006 .

[13]  Tim Schmitz,et al.  Mechanics Of Composite Materials , 2016 .

[14]  Erasmo Carrera,et al.  Component-wise analysis of laminated anisotropic composites , 2012 .

[15]  T. K. Varadan,et al.  Bending of laminated orthotropic cylindrical shells—An elasticity approach , 1991 .

[16]  Ahmed K. Noor,et al.  Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams , 1981 .

[17]  D. J. Dawe,et al.  Numerical studies using circular arch finite elements , 1974 .

[18]  C. Chinosi,et al.  MITC9 Shell Elements Based on Refined Theories for the Analysis of Isotropic Cylindrical Structures , 2013 .

[19]  António J.M. Ferreira,et al.  Analysis of Composite Plates Using a Layerwise Theory and Multiquadrics Discretization , 2005 .

[20]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[21]  A. Love.  I. The small free vibrations and deformation of a thin elastic shell , 1888, Proceedings of the Royal Society of London.

[22]  M. Iura,et al.  A consistent theory of finite stretches and finite rotations, in space-curved beams of arbitrary cross-section , 2001 .

[23]  Erasmo Carrera,et al.  Hierarchical theories of structures based on Legendre polynomial expansions with finite element applications , 2017 .

[24]  D. G. Ashwell,et al.  Finite elements for thin shells and curved members , 1976 .

[25]  As Kaddour,et al.  Input data for test cases used in benchmarking triaxial failure theories of composites , 2012 .

[26]  N. Pagano,et al.  Exact Solutions for Composite Laminates in Cylindrical Bending , 1969 .

[27]  K. Park,et al.  A Curved C0 Shell Element Based on Assumed Natural-Coordinate Strains , 1986 .

[28]  Erasmo Carrera,et al.  Finite Element Analysis of Structures through Unified Formulation , 2014 .

[29]  M. L. Bucalém,et al.  Higher‐order MITC general shell elements , 1993 .

[30]  D. Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals , 2003 .

[31]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

[32]  Masoud Tahani,et al.  Analysis of laminated composite beams using layerwise displacement theories , 2007 .

[33]  Ghodrat Karami,et al.  Finite element analysis of curved beams on elastic foundations , 1989 .

[34]  Erasmo Carrera,et al.  Analysis of laminated beams via Unified Formulation and Legendre polynomial expansions , 2016 .

[35]  Gangan Prathap,et al.  A linear thick curved beam element , 1986 .

[36]  K. Washizu,et al.  Some Considerations on a Naturally Curved and Twisted Slender Beam , 1964 .

[37]  T. Belytschko,et al.  Membrane Locking and Reduced Integration for Curved Elements , 1982 .

[38]  Mohamad S. Qatu,et al.  Vibrations of straight and curved composite beams: A review , 2013 .

[39]  Karan S. Surana,et al.  Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites , 1990 .

[40]  R. P. Shimpi,et al.  A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams , 2001 .

[41]  J. Barlow,et al.  Optimal stress locations in finite element models , 1976 .

[42]  Rakesh K. Kapania,et al.  Recent advances in analysis of laminated beams and plates. Part I - Sheareffects and buckling. , 1989 .

[43]  Erasmo Carrera,et al.  Recent developments on refined theories for beams with applications , 2015 .

[44]  Richard H. Macneal,et al.  Derivation of element stiffness matrices by assumed strain distributions , 1982 .

[45]  Clifford Ambrose Truesdell,et al.  Exact theory of stress and strain in rods and shells , 1957 .

[46]  A. E. H. Love,et al.  The Small Free Vibrations and Deformation of a Thin Elastic Shell , 1887 .