Nonlinear analysis of tape springs: Comparison of twogeometrically exact finiteelementformulations

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Harvey Jenkins
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of Aerospace and Mechanical Engineering, University of Liège, Belgium 11th World

1 Nonlinear analysis of tape springs: Comparison of twogeometrically exact finiteelementformulations Florence Dewalque, Valentin Sonneville and Olivier Brüls Dept. of Aerospace and Mechanical Engineering, University of Liège, Belgium 11th World Congress on Computational Mechanics 5th European Conference on Computational Mechanics Barcelona, July 22, 2014

2 Outline Tape springs Motivations for a formulation of shellson the special Euclidean group Comparison with a classical formulation Test cases Tape springs Conclusions

by its elastic deformation General characteristics: Elastic

3 Tape springs Main features Definition: Thin plate curved along its width used as a compliant mechanism characterised by its elastic deformation General characteristics: Elastic energy Deformation No external energy sources Space applications

4 Tape springs Mechanical behaviour Nonlinear behaviour Buckling, hysteresis and self-locking phenomena Senses of bending Opposite sense of bending 500 Bending moment [Nmm] Equal sense of bending Bending angle [deg]

5 Formulation of shells on SE(3) Motivations: Framework basedon the Lie group theorywhererotations and translations are treated in a unified and frame invariant way Equilibrium equations formulated in a parameterization-free way Singularities due to rotation parameterization naturally avoided Significant reduction of the geometrical nonlinearities Locking-freeand couplednonlinearinterpolation fieldfor translations and rotations No need to update the tangent stiffness matrix at each iteration/time step

6 Classicalformulation = use of the commercial software SAMCEF in which shells are based on the Mindlin-Reissner model Flat plate witha lumpedmass submittedto bending (10 Nm) Mean number of N-R iterations [-] Lie (non constant tan. stiff. matrix) Lie (constant tan. stiff. matrix) Tip displacement [mm] Lie

7 Classicalformulation = use of the commercial software SAMCEF in which shells are based on the Mindlin-Reissner model Flat plate witha lumpedmass submittedto bending (10 Nm) SE(3) Unit line Relative error in SE(3) [%] Nbr. of elements [-]

Flat plate witha lumpedmass submittedto bending (500 Nm) 78.

8 Flat plate witha lumpedmass submittedto bending (500 Nm) Tip displacement [mm] Lie

9 Flat plate witha lumpedmass submittedto bending (500 Nm) Mean number of N-R iterations Lie (non constant tan. stiff. matrix) Tip displacement [mm] Lie

Flat plate submitted to a surface force (1000 N/m²) 31.

10 Flat plate submitted to a surface force (1000 N/m²) Mean number of N-R iterations Tip displacement [mm] Lie (non constant tan. stiff. matrix) Lie

11 Circular plate submitted to a surface force (100 N/m²) Mean number of N-R iterations Central displacement [mm] Lie (non constant tan. stiff. matrix) Lie (constant tan. stiff. matrix) SE(3) Theory 10

12 Square plate submitted to a surface force (1000 N/m²) Mean number of N-R iterations Lie (non constant tan. stiff. matrix) Lie (constant tan. stiff. matrix) Central displacement [mm] Lie

13 Rhombic plate submitted to a surface force (1 N/m²) Lie Theory Mean number of N-R iterations [ Central displacement [mm] Lie (non constant tan. stiff. matrix) Lie (constant tan. stiff. matrix)

14 Barrel roof submittedto a surface force (6250 N/m²) 100 Vertical displacement of B [mm] 10 Vertical displacement of C [mm] Nbr. of elements [-] Lie Nbr. of elements [-] Lie

15 Barrel roof submittedto a surface force (6250 N/m²) Mean number of N-R iterations [-] Lie (non constant tan. stiff. matrix) Lie (constant tan. stiff. matrix) Nbr. of elements [-]

16 Tape springsubmittedto a surface force (00 N/m²) Mean number of N-R iterations Lie (non constant tan. stiff. matrix) Nbr. of elements [-] Tip displacement [mm] Nbr. of elements [-] Lie

17 Conclusions Convergence to the sameresultsfor the formulation on SE(3) and the classical formulation () No needto alwaysupdate the tangent stiffnessmatrix Reduction of the amount of geometric nonlinearities Good representation of nonlinear behaviours Good representationof structures withan initial curvature(tape springs) Next developments: Improvement of the convergence rate Dynamic formulation Adda continuation methodto model the bucklingin tape springs

18 Thankyoufor yourattention Nonlinear analysis of tape springs: Comparison of two geometrically exact finite element formulations Florence Dewalque, Valentin Sonneville and Olivier Brüls Dept. of Aerospace and Mechanical Engineering, University of Liège, Belgium 11th World Congress on Computational Mechanics 5th European Conference on Computational Mechanics Barcelona, July 22, 2014

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