Finite Elements in Analysis and Design

A new simple one-dimensional two-node layered beam element with only 2 degrees of freedom per node is developed for finite element analyses of isotropic and composite beams in this paper. Timoshenko’s composite beam functions are modified and employed to represent the displacement functions of the element, thus providing a unified formulation for slender-to-moderately deep beam analyses, and the notorious shear-locking problem is avoided without employment of remedy schemes. The proposed beam element is demonstrated to be convergent, accurate and computationally efficient for analysis of isotropic and composite beams.

[1]  Pierre Beckers,et al.  Dual analysis with general boundary conditions , 1995 .

[2]  A. Rietz.  Sufficiency of a finite exponent in SIMP (power law) methods , 2001 .

[3]  T. Liszka,et al.  hp-Meshless cloud method , 1996 .

[4]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[5]  Géza Seriani,et al.  Spectral element method for acoustic wave simulation in heterogeneous media , 1994 .

[6]  Jan Drewes Achenbach,et al.  A Strip Element Method for Stress Analysis of Anisotropic Linearly Elastic Solids , 1994 .

[7]  James F. Doyle,et al.  Wave Propagation in Structures , 1989 .

[8]  Fuh-Gwo Yuan,et al.  A higher order finite element for laminated beams , 1990 .

[9]  Gui-Rong Liu,et al.  A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis , 2007 .

[10]  Srinivasan Gopalakrishnan,et al.  Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method , 2006 .

[11]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[12]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[13]  Adnan Ibrahimbegovic,et al.  Plate quadrilateral finite element with incompatible modes , 1992 .

[14]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[15]  Tarun Kant,et al.  New theories for symmetric/unsymmetric composite and sandwich beams with C0 finite elements , 1993 .

[16]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[17]  John Argyris,et al.  BEC: A 2-node fast converging shear-deformable isotropic and composite beam element based on 6 rigid-body and 6 straining modes , 1998 .

[18]  P. Subramanian,et al.  Flexural analysis of laminated composite plates , 1999 .

[19]  Josep Sarrate,et al.  Adaptive finite element strategies based on error assessment , 1999 .

[20]  J. Reddy,et al.  A higher order beam finite element for bending and vibration problems , 1988 .

[21]  Usik Lee,et al.  Spectral-Element Method for Levy-Type Plates Subject to Dynamic Loads , 1999 .

[22]  Eugenio Oñate,et al.  A finite point method for elasticity problems , 2001 .

[23]  N. Aluru.  A point collocation method based on reproducing kernel approximations , 2000 .

[24]  D. Roy Mahapatra,et al.  A spectral finite element model for analysis of axial–flexural–shear coupled wave propagation in laminated composite beams , 2003 .

[25]  K. Y. Lam,et al.  Radial point interpolation based finite difference method for mechanics problems , 2006 .

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

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

[28]  Oden,et al.  An h-p adaptive method using clouds , 1996 .

[29]  Marek Krawczuk,et al.  Modelling of wave propagation in composite plates using the time domain spectral element method , 2007 .

[30]  Irwan Katili,et al.  On a simple triangular reissner/mindlin plate element based on incompatible modes and discrete constraints , 1992 .

[31]  Gang Liu,et al.  Design optimization of stiffened storage tank for spacecraft , 2008 .

[32]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

[33]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[34]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[35]  H. Fukunaga,et al.  Damage Identification of Metallic Structures Using A0 Mode of Lamb Waves , 2008 .

[36]  Zengjie Ge,et al.  A refined discrete triangular Mindlin element for laminated composite plates , 2002 .

[37]  K. P. Rao,et al.  Finite element analysis of laminated anisotropic beams of bimodulus materials , 1984 .

[38]  Moshe Eisenberger,et al.  An exact high order beam element , 2003 .

[39]  Chi King Lee,et al.  On error estimation and adaptive refinement for element free Galerkin method: Part II: adaptive refinement , 2004 .

[40]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[41]  Adnan Ibrahimbegovic,et al.  Quadrilateral finite elements for analysis of thick and thin plates , 1993 .

[42]  Junji Takatsubo,et al.  An NDT technique for composite structures using visualized Lamb-wave propagation , 2007 .

[43]  Gui-Rong Liu,et al.  A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis , 2006 .

[44]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[45]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[46]  Samuel M. Howard,et al.  Dynamic Response and Wave Propagation in Plane Trusses and Frames , 1999 .

[47]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[48]  J. Reddy.  ON LOCKING-FREE SHEAR DEFORMABLE BEAM FINITE ELEMENTS , 1997 .

[49]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[50]  Guirong Liu,et al.  Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM) , 2008 .

[51]  Robert L. Taylor,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part II—A simple triangle , 1993 .

[52]  Peter Cawley,et al.  Optimization of lamb wave inspection techniques , 1992 .

[53]  P. Raveendranath,et al.  A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations , 2001 .

[54]  Ning Hu,et al.  Analysis of Wave Propagation in Beams With Transverse and Lateral Cracks Using a Weakly Formulated Spectral Method , 2007 .

[55]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[56]  Yongsik Kim,et al.  Point collocation methods using the fast moving least‐square reproducing kernel approximation , 2003 .

[57]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[58]  Eugenio Oñate,et al.  A general methodology for deriving shear constrained Reissner‐Mindlin plate elements , 1992 .

[59]  A. Mikkola,et al.  A new locking-free shear deformable finite element based on absolute nodal coordinates , 2007 .

[60]  Carlos E. S. Cesnik,et al.  On Timoshenko-like modeling of initially curved and twisted composite beams , 2002 .

[61]  T. Liszka,et al.  The finite difference method at arbitrary irregular grids and its application in applied mechanics , 1980 .

[62]  Song Cen,et al.  A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates , 2002 .

[63]  Aki Mikkola,et al.  A Linear Beam Finite Element Based on the Absolute Nodal Coordinate Formulation , 2005 .

[64]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[65]  Chen Wanji,et al.  Refined 9‐Dof triangular Mindlin plate elements , 2001 .

[66]  Heung-Jin Chung,et al.  Adaptive nodal generation with the element-free Galerkin method , 2000 .

[67]  Kurt Maute,et al.  Integrated Multidisciplinary Topology Optimization Approach to Adaptive Wing Design , 2006 .

[68]  B. Bourdin.  Filters in topology optimization , 2001 .

[69]  K. Y. Lam,et al.  On efficient finite element modeling of composite beams and plates using higher-order theories and an accurate composite beam element , 1998 .

[70]  K. Maute,et al.  Conceptual design of aeroelastic structures by topology optimization , 2004 .

[71]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[72]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[73]  Sitikantha Roy,et al.  Two finite elements for general composite beams with piezoelectric actuators and sensors , 2009 .

[74]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[75]  K. Y. Dai,et al.  A LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS , 2005 .

[76]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[77]  M. Koshiba,et al.  Finite-Element Analysis of Lamb Wave Scattering in an Elastic Plate Waveguide , 1984, IEEE Transactions on Sonics and Ultrasonics.

[78]  S. Patankar.  Numerical Heat Transfer and Fluid Flow , 2018, Lecture Notes in Mechanical Engineering.

[79]  Do Wan Kim,et al.  Maximum principle and convergence analysis for the meshfree point collocation method , 2006, SIAM J. Numer. Anal..

[80]  Sang-Hoon Park,et al.  A posteriori error estimates and an adaptive scheme of least‐squares meshfree method , 2003 .

[81]  Niels Olhoff,et al.  Topology optimization of continuum structures: A review* , 2001 .

[82]  P. Subramanian,et al.  Flexural analysis of symmetric laminated composite beams using C1 finite element , 2001 .

[83]  K. Svanberg,et al.  An alternative interpolation scheme for minimum compliance topology optimization , 2001 .

[84]  K. S. Kim,et al.  Two simple and efficient displacement‐based quadrilateral elements for the analysis of composite laminated plates , 2004 .

[85]  Chen Wanji,et al.  Refined quadrilateral element based on Mindlin/Reissner plate theory , 2000 .

[86]  Ertugrul Taciroglu,et al.  Much ado about shear correction factors in Timoshenko beam theory , 2010 .

[87]  E. R. A. Oliveira.  Theoretical foundations of the finite element method , 1968 .

[88]  Srinivasan Gopalakrishnan,et al.  A refined higher order finite element for asymmetric composite beams , 2005 .

[89]  Géza Seriani,et al.  Double-grid Chebyshev spectral elements for acoustic wave modeling , 2004 .

[90]  Yi Min Xie,et al.  Evolutionary Structural Optimization , 1997 .

[91]  A. Tucker,et al.  Topology Optimisation of Aircraft Wing Box Ribs , 2004 .

[92]  P. Lardeur,et al.  A discrete shear triangular nine D.O.F. element for the analysis of thick to very thin plates , 1989 .

[93]  G. Y. Li,et al.  A gradient smoothing method (GSM) with directional correction for solid mechanics problems , 2007 .

[94]  Y. Zhu,et al.  A new shear-flexible FRP-reinforced concrete slab element , 2010 .

[95]  Irwan Katili,et al.  A new discrete Kirchhoff-Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields—part II: An extended DKQ element for thick-plate bending analysis , 1993 .

[96]  Hao Yu.  A higher-order finite element for analysis of composite laminated structures , 1994 .

[97]  Eduardo N. Dvorkin,et al.  A formulation of general shell elements—the use of mixed interpolation of tensorial components† , 1986 .

[98]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

[99]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[100]  T. Kundu,et al.  Efficient use of Lamb modes for detecting defects in large plates , 1998 .

[101]  Silvana M. B. Afonso,et al.  Structural optimization strategies for simple and integrally stiffened plates and shells , 2005 .

[102]  Santiago Falcón,et al.  An error indicator for the element free Galerkin method , 2001 .

[103]  Frederic Ward Williams,et al.  EXACT DYNAMIC STIFFNESS MATRIX FOR COMPOSITE TIMOSHENKO BEAMS WITH APPLICATIONS , 1996 .

[104]  Mark A. Bradford,et al.  A layered shear-flexural plate/shell element using Timoshenko beam functions for nonlinear analysis of reinforced concrete plates , 2007 .

[105]  António A. Fernandes,et al.  Modelling of concrete beams reinforced with FRP re-bars , 2001 .

[106]  O. C. Zienkiewicz,et al.  Plate bending elements with discrete constraints: New triangular elements , 1990 .

[107]  Gui-Rong Liu,et al.  The Finite Element Method , 2007 .

[108]  Guirong Liu,et al.  A superconvergent point interpolation method (SC‐PIM) with piecewise linear strain field using triangular mesh , 2009 .

[109]  J. Z. Zhu,et al.  The finite element method , 1977 .