We propose a non-material finite element scheme for modelling large deformations of a closed flexible rod supported by two rigid pulleys in the field of gravity. The mixed Eulerian-Lagrangian kinematic description of circumferential and transverse displacements is beneficial for simulations of moving belt drives. The necessary C 1 inter-element continuity in a compound coordinate system with Cartesian and polar domains requires a nonlinear finite element approximation. The theoretically predicted singular reaction force distribution prevents us from using the technique of Lagrange multipliers for normal contact. A novel semi-analytical solution of the static problem based on the integration of the equations of the nonlinear theory of rods in the free spans as well as in the segments of contact with pulleys is presented for the sake of validation. We demonstrate the mutual convergence of simulation results for a benchmark problem and additionally justify them by comparison against conventional Lagrangian finite element solutions.

We consider an initially horizontal curved elastic strip, which bends and twists under the action of the varying length of the span between the clamped ends and of the gravity force. Equations of the theory of rods, linearized in the vicinity of a largely pre-deformed state, allow for semi-analytical (or sometimes closedform) solutions. A nonlinear boundary value problem determines the vertical bending of a perfect beam, while the small natural curvature additionally leads to torsion and out-of-plane deflections described by the linear equations of the incremental theory. Numerical experiments demonstrate perfect correspondence to the finite element rod model of the strip. Comparisons to the predictions of the shell model allow estimating the range of applicability of the three-dimensional theory of rods. Practically relevant conclusions follow for the case of high pre-tension of the strip.

Studying the mechanics of thin, axially moving strings, beams or plates (e.g.: belt drives, cable cars, . . . ) at mixed Eulerian-Lagrangian description, which features the transformation of material coordinates to spatial ones, is more appropriate than the classical material (Lagrangian) one, see [1,2]. Aiming at testing a newly proposed non-material finite element formulation we study the statics of a looped belt as a rod hanging in contact with two pulleys. Numerical experiments demonstrate rapid mesh convergence as well as correspondence to semi-analytical results and conventional Lagrangian finite element solutions.This is an open access article under the terms of the Creative Commons Attribution-Non-Commercial-NoDerivs Licence 4.0, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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A thin linear elastic strip is clamped at both ends and subjected to a linear stress distribution across its width. We use Kirchhoff beam theory to study this problem. If displacements out of the strips own plane are prohibited, the straight configuration remains stable as long as the compression is not too high. With the three‐dimensional spatial description of the rod theory, we find possible buckling modes even in the case of average tensile stresses in the beam. Comparison with shell and beam finite elements shows excellent agreement with the analytical investigation, also with respect to the supercritical behavior.

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Funding informationÖsterreichische Forschungsförderungsgesellschaft, Grant/Award Number: 861493 A novel mixed Eulerian-Lagrangian rod finite element formulation based on the theory of unshearable, extensible rods is presented. Geometric imperfections (natural curvature) are introduced to investigate and predict the phenomenon of lateral run-off in the benchmark problem of a two-pulley belt drive. The concise description of normal and tangential contact between pulleys and belt contributes to the efficient computation of transient solutions. In resemblance of belt shear theory, it relies on the notion of a thin intermediate layer that connects the belt and the pulleys elastically. Various practically relevant transient solutions are validated against corresponding shell finite element computations available in the literature.

The equations describing torsion of prismatic bars with thin-walled closed cross sections, known as Bredt's formulas, are verified using the method of asymptotic splitting. In particular, the strong formulation of the Saint Venant problem of a straight beam is expanded asymptotically. At first, well-known technical assumptions of the shear stress distribution are validated. Further, the influence of a transverse force acting on the beam is considered. This shear force causes a deformation of the cross section and therefore an adaption of Bredt's formulas. Two distinct formulations of the shear center, called the kinematic and the energetic shear center, are obtained. The latter are verified in numerical experiments.This is an open access article under the terms of the Creative Commons Attribution-Non-Commercial-NoDerivs Licence 4.0, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.