COMPLAS 2023

Keynote

Differential Geometric Split of the Strain Energy of Proportionally Loaded Structures

  • Aminbaghai, Mehdi (Vienna University of Technology)
  • Kalliauer, Johannes (MIT - Massachusetts Institute of Technology)
  • Wagner, Antonia (Vienna University of Technology)
  • Mang, Herbert (Vienna University of Technology)

Please login to view abstract download link

Stretching is generally the preferred mode of deformation of structures. Hence, quantification of its contribution to the strain energy is not only a topic of fundamental research but also one of practical interest. The task of this work is direct computation of the ratio U_M / U where U denotes the strain energy and U_M stands for the contribution of bending and torsion to U. If the contribution of shear to U is negligible, 1 − (U_M / U) is approximately equal to the percentage contribution of stretching to the strain energy. U_M / U is hypothetically set equal to the radius of curvature, ρ, of a curve located on the surface of a unit hypersphere. The curve is described by the vertex of a vector which is obtained from the relevant eigenvector of a linear eigenvalue problem with two indefinite coefficient matrices. They are established with hybrid finite elements, available in a commercial FE program. Their indefiniteness allows for conjugate complex eigenvalues. This enables determination of the load level at which the stiffness of a proportionally loaded structure becomes a minimum. For variable ρ, the aforementioned unit vector follows from normalization of the modified relevant eigenvector, the components of which have been made dimensionless. The task of the numerical investigation is to verify the hypothesis U_M / U = ρ. The first two examples refer to the limiting cases of pure stretching and pure bending. The other two are characterized by a maximum and a minimum value, respectively, of the stiffness of the structure concerned. However, their positions do not coincide with the ones of the máximum values of U_M / U of the two structures.