Article Contents
Article Contents

# Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

• * Corresponding author: Bouazza Braikat
• 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.

 Citation:

• Figure 1.  Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour

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 5.  Geometrical characteristics of sections $A$ and $B$

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 $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$

Table 1.  Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step

 Solvers $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $10$ $-3.73$ $9$ $-3.71$ $8$ $-3.71$ $2\, 10^{-3}$ $12$ $-3.72$ $10$ $-3.71$ $9$ $-3.70$ $3\, 10^{-3}$ $13$ $-3.69$ $11$ $-3.67$ $10$ $-3.65$

Table 2.  Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order

 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $7$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $8$ $a_{max}<1$ $a_{max}<1$ $2995$ $32000$ $3890$ $9$ $a_{max}<1$ $2810$ $36000$ $4252$ $2711$ $36000$ $4102$ $10$ 2850 40000 4900 $2600$ $40000$ $4470$ $2480$ $40000$ $4262$ $15$ 630 60000 12376 $612$ $60000$ $12023$ $520$ $60000$ $10210$ $20$ $320$ $80000$ $25896$ $309$ $80000$ $25000$ $280$ $80000$ $22640$

Table 3.  Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step

 Solver $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $6$ $-5.23$ $4$ $-5.2$ $3$ $-5.13$ $2\, 10^{-3}$ $12$ $-5.10$ $9$ $-4.80$ $7$ $-4.62$ $3\, 10^{-3}$ $14$ $-4.91$ $11$ $-4.79$ $8$ $-4.60$ $4\, 10^{-3}$ $15$ $-4.88$ $12$ $-4.70$ $10$ $-4.55$

Table 4.  Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order

 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $2$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $3$ $a_{max}<1$ $a_{max}<1$ $12$ $600$ $26$ $4$ $a_{max}<1$ $13$ $800$ $42$ $11$ $800$ $30$ $5$ $a_{max}<1$ $8$ $1000$ $46$ $6$ $1000$ $34$ $6$ $15$ $1200$ $102$ $7$ $1200$ $54$ $5$ $1200$ $36$ $7$ $4$ $1400$ $126$ $2$ $1400$ $65$ $1$ $1400$ $40$
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Tables(4)