Figure 1.
Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour
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2019, 12(6): 1685-1708.
doi: 10.3934/dcdss.2019113
Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings
| 1. | Laboratoire d'Ingénierie et Matériaux (LIMAT), Faculté des Sciences Ben M'Sik, Hassan Ⅱ University of Casablanca, Avenue Cdt Driss El Harti B.P. 7955, Sidi Othman, Casablanca, Morocco |
| 2. | Laboratoire de Génie Mécanique (LGM), Faculté des Sciences et Techniques, Fès, Université Sidi Mohamed Ben Abdellah, Route d'Imouzzer B.P. 2202, Fès, Maroc |
| 3. | Laboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3), Université de Lorraine, Metz, CNRS UMR 7239, Ile du Saulcy, 57057, France |
This paper aims to investigate, in large displacement and torsion context, the nonlinear dynamic behavior of thin-walled beams with open cross section subjected to various loadings by high-order implicit solvers. These homotopy transformations consist to modify the nonlinear discretized dynamic problem by introducing an arbitrary invertible pre-conditioner $ [K^\star] $ and an arbitrary path following parameter. The nonlinear strongly coupled equations of these structures are derived by using a $ 3D $ nonlinear dynamic model which accounts for large displacements and large torsion without any assumption on torsion angle amplitude. Coupling complex structural phenomena such that warping, bending-bending, and flexural-torsion are taken into account.
Two examples of great practical interest of nonlinear dynamic problems of various thin-walled beams with open section are presented to validate the efficiency and accuracy of high-order implicit solvers. The obtained results show that the proposed homotopy transformations reveal a few number of matrix triangulations. A comparison with Abaqus code is presented.
Keywords:
Thin-walled beam,
open cross section,
large displacement,
large torsion,
forced nonlinear dynamic,
finite element,
homotopy,
power series expansion,
Asymptotic Numerical Method (ANM).
Citation:
Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri, Noureddine Damil. Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings. Discrete & Continuous Dynamical Systems - S,
2019, 12
(6)
: 1685-1708.
doi: 10.3934/dcdss.2019113
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References:
| [1] |
Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar |
| [2] |
E. L. Allgower and K. Georg,
Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990.
doi: 10.1007/978-3-642-61257-2. |
| [3] |
R. D. Ambrosini, J. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885. Google Scholar |
| [4] |
K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar |
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J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar |
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K. Behdinan, M. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161. Google Scholar |
| [7] |
P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59. Google Scholar |
| [8] |
O. Bourihane, B. Braikat, M. Jamal, F. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146. Google Scholar |
| [9] |
S. Boutmir, B. Braikat, M. Jamal, N. Damil, B. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460. Google Scholar |
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M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar |
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E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139. Google Scholar |
| [12] |
A. Ed-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255. Google Scholar |
| [13] |
Y. Guevel, G. Girault and J. M. Cadou,
Numerical comparisons of high-order nonlinear solvers for the transient $ \textbf{N} $avier-$ \textbf{S} $tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.
doi: 10.1016/j.cam.2014.12.008. |
| [14] |
D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112. Google Scholar |
| [15] |
M. Jamal, B. Braikat, S. Boutmir, N. Damil and M. Potier-Ferry,
A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.
doi: 10.1007/s00466-002-0301-7. |
| [16] |
T. N. Le, J. M. Battini and M. Hjiaj,
Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.
doi: 10.1007/s00466-011-0585-6. |
| [17] |
T. N. Le, J. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127. Google Scholar |
| [18] |
S. Mesmoudi, A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11. Google Scholar |
| [19] |
F. Mohri, N. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683. Google Scholar |
| [20] |
F. Mohri, L. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446. Google Scholar |
| [21] |
F. Mohri, N. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315. Google Scholar |
| [22] |
F. Mohri, N. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683. Google Scholar |
| [23] |
F. Mohri, A. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990. Google Scholar |
| [24] |
H. Mottaqui, B. Braikat and N. Damil,
Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.
doi: 10.1016/j.cma.2010.01.020. |
| [25] |
H. Mottaqui, B. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22. Google Scholar |
| [26] |
N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94. Google Scholar |
| [27] |
E. J. Sapountzakis and I. C. Dikaros,
Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.
doi: 10.1007/s11071-013-0779-x. |
| [28] |
A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380. Google Scholar |
| [29] |
V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar |
| [30] |
O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar |
show all references
References:
| [1] |
Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar |
| [2] |
E. L. Allgower and K. Georg,
Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990.
doi: 10.1007/978-3-642-61257-2. |
| [3] |
R. D. Ambrosini, J. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885. Google Scholar |
| [4] |
K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar |
| [5] |
J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar |
| [6] |
K. Behdinan, M. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161. Google Scholar |
| [7] |
P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59. Google Scholar |
| [8] |
O. Bourihane, B. Braikat, M. Jamal, F. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146. Google Scholar |
| [9] |
S. Boutmir, B. Braikat, M. Jamal, N. Damil, B. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460. Google Scholar |
| [10] |
M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar |
| [11] |
E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139. Google Scholar |
| [12] |
A. Ed-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255. Google Scholar |
| [13] |
Y. Guevel, G. Girault and J. M. Cadou,
Numerical comparisons of high-order nonlinear solvers for the transient $ \textbf{N} $avier-$ \textbf{S} $tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.
doi: 10.1016/j.cam.2014.12.008. |
| [14] |
D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112. Google Scholar |
| [15] |
M. Jamal, B. Braikat, S. Boutmir, N. Damil and M. Potier-Ferry,
A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.
doi: 10.1007/s00466-002-0301-7. |
| [16] |
T. N. Le, J. M. Battini and M. Hjiaj,
Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.
doi: 10.1007/s00466-011-0585-6. |
| [17] |
T. N. Le, J. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127. Google Scholar |
| [18] |
S. Mesmoudi, A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11. Google Scholar |
| [19] |
F. Mohri, N. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683. Google Scholar |
| [20] |
F. Mohri, L. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446. Google Scholar |
| [21] |
F. Mohri, N. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315. Google Scholar |
| [22] |
F. Mohri, N. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683. Google Scholar |
| [23] |
F. Mohri, A. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990. Google Scholar |
| [24] |
H. Mottaqui, B. Braikat and N. Damil,
Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.
doi: 10.1016/j.cma.2010.01.020. |
| [25] |
H. Mottaqui, B. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22. Google Scholar |
| [26] |
N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94. Google Scholar |
| [27] |
E. J. Sapountzakis and I. C. Dikaros,
Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.
doi: 10.1007/s11071-013-0779-x. |
| [28] |
A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380. Google Scholar |
| [29] |
V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar |
| [30] |
O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar |
Figure 2.
Axial force $N$ , bending moments $M_{y}$ and $M_{z}$ , bimoment $B_{\omega}$ and St-Venant torsion moment $M_{sv}$
Figure 3.
Section beam under concentrated and distributed forces
Figure 4.
External dynamical loading and its time evolution applied on the U-mono-symmetrical thin-walled beam with open cross section
Figure 6.
Response curves obtained by the high-order implicit solver $Alg_3$ and by Abaqus code, Time evolution of displacement components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
Figure 7.
Cantilever bi-symmetrical beam with steel I cross section under eccentric loading and its time evolution
Figure 8.
Thin-walled beam with steel I cross section under transverse eccentric force $F_{z}(t)$ and its point of application
Figure 9.
Response curves obtained by the high-order implicit solver $Alg_3$ , by Abaqus code and by Sapountzakis: Time evolution of components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
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