Finite Element Method Application For Nonlinear Mechanical Response Of Three-Dimensonal Beams Using Mathematica
Abstract
Mechanical behavior of beams under large rotations and
displacements was investigated. Using co-rotational approach given by
Crisfield, three dimensional beam finite elements was modeled in Mathematica
environment. The symbolic process of Mathematica provides keeping the
parameters in the model as variables and beams having different geometries can
be modelled easily. The yielding non-linear equation system was solved by
utilizing Newton-Raphson technique. Dynamic balance equations and numerical
time integration method were introduced, the solution process was left as a
future work, however. Comparisons with the results of ANSYS and those of the
works from the literature are accomplished.
References
- [1] Argyris J., “An excursion into large rotations”, Comp. Methods Appl. Mech. Engrg., 32: 85–155, (1982).[2] Oran C. and Kassýmali A., “Large deformations of framed structures under static and dynamic loads”, Computers & Structures, 6: 539–547, (1976).[3] Belytschko T., “Large displacement, transient analysis of space frames”, Int. J. for Num. Meth. in Engrg., 11: 65–84, (1977).[4] Crisfield M.A., “A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements”, Comp. Methods Appl. Mech. Engrg., 81: 131–150, (1990).[5] Hsiao K.M., Lin J., Lin W.Y., “A consistent co-rotational finite element formulation for geometrically nonlinear analysis of 3-D beams”, Comp. Methods Appl. Mech. Engrg., 169: 1–18, (1999).[6] Pai P.F., “Large deformation tests and total-Lagrangian finite-element analyses of flexible beams”, Int. J. of Solids and Structures, 37: 2951–2980, (2000).[7] Le T.-N., Battini J.-M., Hijiaj M., “Dynamics of 3D beam elements in a corotational context: A comparative study of established and new formulations”, Finite Elements in Analysis and Design, 61: 97-111, (2012).[8] Jonker J.B., “A geometrically nonlinear formulation of three-dimensional beam element for solving large deflection multibody system problems”, International Journal of Non-Linear Mechanics, 53: 63-74, (2013).[9] Bathe K.-J., “Large displacement analysis of three-dimensional beam structures”, Int. J. For Num. Meth. in Engrg., 14: 961–986, (1979).[10] Simo J.C., “A finite strain beam formulation. The three dimensional dynamic problem, Part-I”, Comp. Methods Appl. Mech. Engrg., 49: 55–70, (1985).[11] Rhim J. and Lee S.W., “A vectorial approach to computational modelling of beams undergoing finite rotations”, Int. J. for Num. Meth. in Engrg., 41: 527–540, (1998).[12] Le T.-N., Battini J.-M., Hijiaj M., “A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures”, Comp. Methods Appl. Mech. Engrg., 269: 538-65, (2014). [13] de Miranda S., Madeo A., Melchionda D., Patruno L. and Ruggerini A.W., “A corotational based geometrically nonlinear Generalized Beam Theory: buckling FE analysis”, Int. J. Solids and Structures, 121: 212-227, (2017).[14] Cho H., Kim H., Shin S., “Geometrically nonlinear dynamic formulation for three-dimensional co-rotational solid elements”, Comput. Meth. Appl. Mech. Engrg., 323: 301-320, (2018).
Üç Boyutlu Kirişlerin Doğrusal Olmayan Mekanik Davranışı için Mathematica Kullanarak Sonlu Elemanlar Yöntemi Uygulaması
Abstract
Kirişlerin büyük
dönmeler ve yer değiştirmeler altında mekanik davranışı incelendi. Crisfield
tarafından verilen eş-dönüşlü formülasyon yardımıyla üç boyutlu kiriş sonlu
elemanları Mathematica yazılımı içerisinde modellendi. Mathematica yazılımının
sembolik işlemcisi sayesinde modele ait parametreler değişken tutulabilir ve
değişik geometrilerdeki kirişler kolayca modellenebilir. Doğrusal olmayan
denklem takımları Newton-Raphson yöntemi ile çözüldü. Dinamik hareket denklemi
ve sayısal zaman integrasyon yöntemi ortaya konuldu yalnız çözümlemeler ileriki
çalışmalara bırakıldı. Literatürdeki çalışmaların ve ANSYS programının
sonuçlarıyla karşılaştırmalar yapıldı.
References
- [1] Argyris J., “An excursion into large rotations”, Comp. Methods Appl. Mech. Engrg., 32: 85–155, (1982).[2] Oran C. and Kassýmali A., “Large deformations of framed structures under static and dynamic loads”, Computers & Structures, 6: 539–547, (1976).[3] Belytschko T., “Large displacement, transient analysis of space frames”, Int. J. for Num. Meth. in Engrg., 11: 65–84, (1977).[4] Crisfield M.A., “A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements”, Comp. Methods Appl. Mech. Engrg., 81: 131–150, (1990).[5] Hsiao K.M., Lin J., Lin W.Y., “A consistent co-rotational finite element formulation for geometrically nonlinear analysis of 3-D beams”, Comp. Methods Appl. Mech. Engrg., 169: 1–18, (1999).[6] Pai P.F., “Large deformation tests and total-Lagrangian finite-element analyses of flexible beams”, Int. J. of Solids and Structures, 37: 2951–2980, (2000).[7] Le T.-N., Battini J.-M., Hijiaj M., “Dynamics of 3D beam elements in a corotational context: A comparative study of established and new formulations”, Finite Elements in Analysis and Design, 61: 97-111, (2012).[8] Jonker J.B., “A geometrically nonlinear formulation of three-dimensional beam element for solving large deflection multibody system problems”, International Journal of Non-Linear Mechanics, 53: 63-74, (2013).[9] Bathe K.-J., “Large displacement analysis of three-dimensional beam structures”, Int. J. For Num. Meth. in Engrg., 14: 961–986, (1979).[10] Simo J.C., “A finite strain beam formulation. The three dimensional dynamic problem, Part-I”, Comp. Methods Appl. Mech. Engrg., 49: 55–70, (1985).[11] Rhim J. and Lee S.W., “A vectorial approach to computational modelling of beams undergoing finite rotations”, Int. J. for Num. Meth. in Engrg., 41: 527–540, (1998).[12] Le T.-N., Battini J.-M., Hijiaj M., “A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures”, Comp. Methods Appl. Mech. Engrg., 269: 538-65, (2014). [13] de Miranda S., Madeo A., Melchionda D., Patruno L. and Ruggerini A.W., “A corotational based geometrically nonlinear Generalized Beam Theory: buckling FE analysis”, Int. J. Solids and Structures, 121: 212-227, (2017).[14] Cho H., Kim H., Shin S., “Geometrically nonlinear dynamic formulation for three-dimensional co-rotational solid elements”, Comput. Meth. Appl. Mech. Engrg., 323: 301-320, (2018).
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Engineering |
| Journal Section | Research Article |
| Authors | |
| Publication Date | December 1, 2019 |
| Submission Date | July 1, 2018 |
| Published in Issue | Year 2019 |
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