r3.08.01 « Exact » beam elements

Summary:

This document shows the Code_Aster beam elements based on an exact resolution of the equations of the continuous model performed for each element in the mesh.

The beams are straight: elements POU_D_T and POU_D_E. The curved beams will be modelled using a sufficiently fine mesh, on which a straight beam model is assigned. The cross section, which is constant or variable over length, may be of any shape. The material is homogeneous, isotropic, and linear elastic.

The hypotheses adopted are as follows:

• Euler hypothesis: transverse shear is neglected, as well as rotational inertia. This hypothesis is verified for strong impulses: element POU_D_E.

• Timoshenko hypothesis: transverse shear and all inertia terms are taken into account. This hypothesis is to be used for low levels of movement: elements POU_D_T.

• Saint-Venant hypothesis: the twist is free.

The treatment of the various loads and the quantities expected as a result (stress-efforts) is also presented.

Foreword

This reference documentation of the beam elements was produced from work carried out by M.T.Bourdeix, P.Hemon, O.Wilk from the Aerotechnical Institute of the National Conservatory of Arts and Crafts, as part of an External Research and Development Contract with this laboratory.

The volume of this document is due both to the precision sought and to the didactic nature of the presentation, which is deliberately preserved.

• 1. Notations
• 2. Introduction
• 3. The equations of motion
  • 3.1. Traction-compression
  • 3.2. The pure twist (Saint-Venant twist)
  • 3.3. Simple bending
• 4. Right beam element
  • 4.1. Longitudinal traction movement - compression
  • 4.2. Free torsional movement around the longitudinal axis
  • 4.3. Flexion movement
  • 4.4. Reduced mass matrix using the concentrated mass technique
  • 4.5. Centrifugal stiffness matrix
• 5. Special straight beams
  • 5.1. Eccentricity of the torsional axis in relation to the neutral axis
  • 5.2. Variable sections
• 6. Geometric stiffness - Prestressed structure
• 7. Big trips
• 8. Curved beam
  • 8.1. Flexibility matrix for bending in the plane of the beam [C1]
  • 8.2. Flexibility matrix for bending out of the plane of the beam [C2]
• 9. Loads
  • 9.1. Deformation loading
  • 9.2. Loading due to gravity
  • 9.3. Distributed loads
  • 9.4. Thermal charging
  • 9.5. Electric charging
• 10. Force twister - Stress twister (or generalized efforts) - Nodal forces and reactions
  • 10.1. The twister of efforts
  • 10.2. The stress tensor
  • 10.3. Calculation of nodal forces and reactions
• 11. Bar element
• 12. Bibliography