Energy-momentum conserving time-stepping algorithms for nonlinear dynamics of planar and spatial Euler-Bernoulli/Timoshenko beams

Large deformations of flexible beams can be described using either the co-rotational approach or the total Lagrangian formalism. The co-rotational method is an attractive approach to derive highly nonlinear beam elements because it combines accuracy with numerical efficiency. On the other hand, the total Lagrangian formalism is the natural setting for the construction of geometrically exact beam theories. Classical time integration methods such as Newmark, standard midpoint rule or the trapezoidal rule do suffer severe shortcomings in nonlinear regimes. The construction of time integration schemes for highly nonlinear problems which conserve the total energy, the momentum and the angular momentum is addressed for planar co-rotational beams and for a geometrically exact spatial Euler-Bernoulli beam. In the first part of the thesis, energy-momentum conserving algorithms are designed for planar co-rotational beams. Both Euler-Bernoulli and Timoshenko kinematics are addressed. These formulations provide us with highly complex non-linear expressions for the internal energy as well as for the kinetic energy which involve second derivatives of the displacement field. The main idea of the algorithm is to circumvent the complexities of the geometric non-linearities by resorting to strain velocities to provide, by means of integration, the expressions for the strain measures themselves. Similarly, the same strategy is applied to the highly nonlinear inertia terms. Several examples have been considered in which it was observed that energy, linear momentum and angular momentum are conserved for both formulations even when considering very large number of time-steps. Next, 2D elasto-(visco)-plastic fiber co-rotational beams element and a planar co-rotational beam with generalized elasto-(visco)-plastic hinges at beam ends have been developed and compared against each other for impact problems. Numerical examples show that strain rate effects influence substantially the structure response. In the second part of this thesis, a geometrically exact 3D Euler-Bernoulli beam theory is developed. The main challenge in defining a three-dimensional Euler-Bernoulli beam theory lies in the fact that there is no natural way of defining a base system at the deformed configuration. A novel methodology to do so leading to the development of a spatial rod formulation which incorporates the Euler-Bernoulli assumption is provided. The approach makes use of Gram-Schmidt orthogonalisation process coupled to a one-parametric rotation to complete the description of the torsional cross sectional rotation and overcomes the non-uniqueness of the Gram-Schmidt procedure. Furthermore, the formulation is extended to the dynamical case and a stable, energy conserving time-stepping algorithm is developed as well. Many examples confirm the power of the formulation and the integration method presented.QC 20181106