r5.03.40 Static and dynamic modeling of beams in large rotations

Summary:

This note gives a mechanical formulation of beams in large displacements and rotations but with elastic behavior. The main difficulty in analyzing rotations lies in the fact that they are not switchable and do not constitute a vector space, but a manifold.

At any moment, the configuration of a straight section of a beam is defined by the displacement vector of its center of gravity and the rotation vector of the system of the main axes of inertia with respect to a reference position. As in classical beam theory, the internal forces are reduced to their resultant and their moment on the line of the centers of the sections. The associated deformations are defined.

The linearization of internal forces with respect to displacements leads to the usual stiffness matrix, which is symmetric, and, because of the large displacements and rotations, to the geometric rigidity matrix, which is arbitrary.

The linearization of inertial forces leads, for translational motion, to the usual mass matrix that is symmetric and, for rotational motion, to a much more complicated matrix with no symmetry at all.

The temporal integration scheme is that of Newmark.

This modeling was tested on five reference problems: three static problems and two dynamic problems.

This work was undertaken as part of the development of modeling tools for line and station components. The aim of modeling is the dynamic study of conductors equipped with spacers (for lines) or descents on equipment (for stations) and subjected to Laplace forces resulting from short circuit currents.

• 1. Notations
• 2. Introduction
• 3. Kinematics of a beam in finite rotations
• 4. Rotation vector and operator
  • 4.1. Vector-rotation
  • 4.2. Rotation operator
• 5. Transition from local to general roads
• 6. Inner forces, deformations and the law of behavior
  • 6.1. Inner efforts
  • 6.2. Variation of curvature at a point on the center line
  • 6.3. Virtual work in the beam and the vector of deformations
  • 6.4. Law of behavior
• 7. Elementary inertial forces
• 8. Equation of motion and calculation process
  • 8.1. Equation of undamped motion
  • 8.2. Process of a calculation
• 9. Linearization of the equations of motion
  • 9.1. Stiffness matrices
  • 9.2. Inertia matrices
• 10. Finite element implementation
  • 10.1. Deformation matrix and internal forces
  • 10.2. Stiffness matrices
  • 10.3. Inertial forces
  • 10.4. Inertia matrix
  • 10.5. External forces given
  • 10.6. Linear iteration system
  • 10.7. Displacement, speed, and acceleration update
  • 10.8. Curvature variation vector update
  • 10.9. Initialization before iterations
• 11. Schematic organization of a calculation
  • 11.1. Static calculation
  • 11.2. Dynamic calculation
• 12. Use by Code_Aster
• 13. Numerical simulations
  • 13.1. Embedded straight beam subjected to a concentrated moment at the end (test case SSNL103)
  • 13.2. Recessed swivel arch loaded at the top
  • 13.3. Circular arc of 45° embedded and subjected at the end to a force perpendicular to its plane
  • 13.4. Movement of a gallows
  • 13.5. Rotation of a robot arm
• 14. Bibliography
• 15. Description of document versions