FEA versus IGA in a two-node beam element based on unified and integrated method. (English) Zbl 1488.74002
Summary: This paper presents a new concept called Unified and Integrated Method for a shear deformable beam element. In this method, Timoshenko beam theory is unified and integrated in such a way that takes into account the effect of transverse shear and maintains the shear locking free condition at the same time to generate proper behavior in the analysis of thin to thick beams. The unified and integrated method is applied to finite element analysis (FEA) and isogeometric analysis (IGA) on two-node beam element. This method will be used to analyze uniformly loaded beams with various boundary conditions. A shear influence factor of \(\phi\), which is a function of beam thickness ratio (\(L/h\)), is expressed explicitly as control of the transverse shear strain effect. The analysis gives interesting results showing that applying the unified and integrated method in FEA and IGA will yield exact values of DOF’s and displacement function even when using only a single element. Numerical examples demonstrate the validity and efficiency of the unified and integrated methods.
MSC:
| 74-10 | Mathematical modeling or simulation for problems pertaining to mechanics of deformable solids |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S22 | Isogeometric methods applied to problems in solid mechanics |
Keywords:
finite element analysis; isogeometric analysis; unified and integrated method; Timoshenko beam theoryReferences:
| [1] | S. TIMOSHENKO, On the correction for shear of differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), pp. 744-746. |
| [2] | S. TIMOSHENKO, On the transverse vibrations of bars of uniform cross section, Philosophical Magazine, 43 (1922), pp. 125-131. |
| [3] | G. PRATHAP AND G. BHASHYAM, Reduced integration and the shear-flexible beam element, Int. J. Numer. Methods Eng., 18 (1982), pp. 172-178. · Zbl 0473.73084 |
| [4] | K.-U. BLETZINGER, M. BISCHOFF AND E. RAMM, A unified approach for shear-locking-free triangular and rectangular shell finite elements, Computers and Structures, 75 (2000), pp. 321-334. |
| [5] | J. L. BATOZ AND G. DHATT, Modélisation des Structures paréléments Finis : Poutres et Plaques, Vol. 2, Hermes Science Publications, 1990. · Zbl 0831.73002 |
| [6] | I. KATILI, Unified and integrated approach in a new Timoshenko beam element, Euro. J. Comput. Mech., 26 (2017), pp. 282-308. |
| [7] | I. KATILI, A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields, part I: An extended DKT element for thick-plate bending analysis, Int. J. Numer. Methods Eng., 36 (1993), pp. 1859-1883. · Zbl 0775.73263 |
| [8] | I. KATILI, 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, Int. J. Numer. Methods Eng., 36 (1993), pp. 1885-1908. · Zbl 0775.73264 |
| [9] | I. KATILI, J. L. BATOZ, I. J. MAKNUN, A. HAMDOUNI AND O. MILLET, The development of DKMQ plate bending element for thick to thin shell analysis based on naghdi/reissner/mindlin shell theory, Finite Elements in Analysis and Design, 100 (2015), pp. 12-27. |
| [10] | I. KATILI, I. J. MAKNUN, A. HAMDOUNI AND O. MILLET, Application of DKMQ element for Composite plate bending structures, Composite Structures, 132 (2015), pp. 166-174. |
| [11] | I. J. MAKNUN, I. KATILI, O. MILLET AND A. HAMDOUNI, Application of DKMQ element for twist of thin-walled beams: comparison with Vlasov theory, Int. J. Comput. Methods Eng. Sci. Mech., 17 (2016), pp. 391-400. · Zbl 07871442 |
| [12] | I. KATILI, J. L. BATOZ, I. J. MAKNUN AND P. LARDEUR, A comparative formulation of DKMQ, DSQ and MITC4 quadrilateral plate elements with new numerical results based on s-norm tests, Computer and Structures, 204 (2018), pp. 48-64. |
| [13] | I. KATILI, I. J. MAKNUN, J. L. BATOZ AND A. IBRAHIMBEGOVIĆ, Shear deformable shell ele-ment DKMQ24 for composite structures, Composite Structures, 202 (2018), pp. 182-200. |
| [14] | I. KATILI, I. J. MAKNUN, J. L. BATOZ AND A. M. KATILI, Asymptotic equivalence of DKMT and MITC3 elements for thick composite plates, Composite Structures, 206 (2018), pp. 363-379. |
| [15] | H. IRPANNI, I. KATILI AND I. J. MAKNUN, Development DKMQ shell element with five degrees of freedom per nodal, Int. J. Mech. Eng. Robot. Res., 6 (2017), pp. 248-252. |
| [16] | I. KATILI, I. J. MAKNUN AND E. TJAHJONO, Alisyahbana I. Error estimation for the DKMQ24 shell element by using various recovery methods, Int. J. Tech., 6 (2017), pp. 1060-1069. |
| [17] | F. T. WONG, ERWIN RICHARD A AND I. KATILI, Development of the DKMQ element for buck-ling analysis of shear-deformable plate bending, Proc. Eng., 171 (2017), pp. 805-812. |
| [18] | I. SENJANOVIĆ, N. VLADIMIR, M. TOMIĆ, An advanced theory of moderately thick plate vibrations, J. Sound Vibration, 332 (2013a), pp. 1868-1880. |
| [19] | H. T. THAI, T. K. NGUYEN, T. P. VO AND T. NGO, A new simple shear deformation plate theory, Composite Structures, 171 (2017), pp. 277-285. |
| [20] | I. KATILI AND R. ARISTIO, Isogeometric Galerkin in rectangular plate bending problem based on UI approach, Euro. J. Mech. A Solids, 67 (2018), pp. 92-107. · Zbl 1406.74440 |
| [21] | J. KIENDL, F. AURICCHIO, T. J. R. HUGHES AND A. REALI, Single-variable formulations and isogeometric discretization for shear deformable beams, Computer Methods Appl. Mech. Eng., 284 (2015), pp. 988-1004. · Zbl 1423.74492 |
| [22] | K. KAPUR, Vibrations of a Timoshenko beam using finite-element approach, J. Acoustic Soc. Am., 40 (1966), pp. 1058-1063. · Zbl 0145.45901 |
| [23] | X. F. LI, A unified approach for analyzing static and dynamic behavior of functionally graded Timo-shenko and Bernoulli-Euler beam, J. Sound Vibration, 318 (2008), pp. 1210-1229. |
| [24] | G. FALSONE AND D. SETTINERI, An Bernoulli-Euler-like finite element method for Timoshenko beams, Mech. Res. Commun., 38 (2011), pp. 12-16. · Zbl 1272.74609 |
| [25] | I. KATILI, T. SYAHRIL AND A. M. KATILI, Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko’s theory, Composite Structures, 242 (2020), https://doi.org/10.1016/j.compstruct.2020.112130. · doi:10.1016/j.compstruct.2020.112130 |
| [26] | C. DE BOOR, A Practical Guide to Splines (revised edition), Springer, 2001. · Zbl 0987.65015 |
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