Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm Oussama Bourihane a , Bouazza Braikat a,⇑ , Mohammad Jamal a , Foudil Mohri b , Noureddine Damil a a Laboratoire d’Ingénierie et Matériaux (LIMAT), Faculté des Sciences Ben M’Sik, Université Hassan II de Casablanca, BP 7955, Sidi Othman, Casablanca, Morocco b Université de Lorraine, Laboratoire d’Étude des Microstructures et de Mécanique des Matériaux (LEM3), CNRS UMR 7239, Ile du Saulcy, 57 057 Metz, France article info Article history: Received 8 December 2015 Revised 2 April 2016 Accepted 5 April 2016 Keywords: Thin-walled beam Open section Nonlinear dynamic Finite element Newmark scheme Homotopy Higher order Asymptotic Numerical Method abstract In this paper, the forced nonlinear dynamic behavior of thin-walled beams with open cross section under external dynamic loads is analyzed by means of a high order implicit algorithm. This algorithm is devel- oped using a 3D nonlinear model that takes into account the large torsion without any assumption on the torsion angle amplitude neither in the constitutive law nor in the derivation for governing dynamic equa- tions. This algorithm is built by employing the following four steps: 1 – the space and time discretization procedures, 2 – a change of variable, 3 – a homotopy transformation, 4 – techniques used in the Asymptotic Numerical Method (ANM) (Cochelin et al., 2007; Mottaqui et al., 2010) [1,2]. The originality of this work reside in the fact that we use, for the first time, this algorithm for nonlinear analysis of thin-walled beams with open cross section under an arbitrary load. The space and time discretizations are performed respectively by the finite elements method and by the classical implicit Newmark scheme. The performance of the high order implicit algorithm is tested on four examples of nonlinear dynamic: a mono-symmetrical beam with a T cross section under external dynamic load, a mono-symmetrical beam with U cross-section under external dynamic load, a bi-symmetrical clamped-free beam IPE300 under harmonic loads and a bi-symmetrical simply supported beam with cruciform section under harmonic loads. A comparison of the obtained results with those computed by the Abaqus industrial code is given. This comparison confirms the robustness, accuracy and efficiency of this high order implicit algorithm. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction Thin-walled beams with open cross sections are commonly used in civil, aeronautic and mechanical engineering structures. Their structural behavior is extremely sensitive to torsional warp- ing and local buckling, particularly when subjected to dynamic loadings at frequencies close to their natural frequencies causing nonlinear vibrations with large deformations. In addition, the cou- pling effects among bending, torsional, shear and axial vibrations must be taken into account in order to understand their dynamic behavior and for their safety design of these structures. Therefore, to capture correctly the coupling between the bend- ing, the twisting, the stretching and the large torsion of thin-walled beams with open sections under external dynamic loadings and predict accurately their dynamic responses, the required formula- tion may be achieved by a nonlinear dynamic model. The nonlinear dynamic of thin-walled beams with open cross section undergoing large deflections has been the subject for several research works. Several co-rotational and lagrangian finite beam formulations have been proposed in the literature [3–11]. Haijuan proposed in [12] a finite element formulation for nonlinear free vibration of thin- walled curved beams with non-symmetric open cross section and adopting a direct iteration technique to solve the nonlinear eigen- value problem. Behiano et al. proposed in [13] a generalized beam theory (GBT) formulation to analyze the local and global dynamic behavior of open section thin-walled members under arbitrary excitation loadings based on the principle of modal superposition. Sapountzkis and Tsipiras presented in [14] a boundary element method (BEM) for evaluating the warping shear stress of bars of arbitrary nonlinear doubly symmetric constant cross-section undergoing nonuniform torsional vibration. A finite element study was devoted in [15] to the nonlinear free torsional vibrations of thin-walled beams with bi-symmetric open cross section only in the elastic case. The solution is obtained by taking into account the linearization of the displacement field. Simo et al. [16] presented a fully nonlinear 3D rod model based on a exact kinematics including the effects of shear and torsional-warping deformations. The solution of this model is obtained by the finite element method. Egidio et al. [17,18] http://dx.doi.org/10.1016/j.engstruct.2016.04.003 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: b.braikat@gmail.com (B. Braikat). Engineering Structures 120 (2016) 133–146 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct