(Created 2011-09-01.)

BEAM THEORY | VSMF15 |

**Aim**

The course shall give knowledge about the action of straight and curved beams and about theories for calculation of stiffness, deformations, stresses and instability of beams loaded in 3D, including influence of eigenstresses, and with cross-sections that may vary along the beam and have arbitrary geometrical shape, including thinn walled cross-sections.

*Knowledge and understanding*

For a passing grade the student must

- Be able to give account of different kinds of beams, their mechanical action and performance, and phenomena that limit their servicabilty.
- Be able to give account of the beam theories of Bernoulli-Euler, Timoshenko, St Venant and Vlasov, and for the basics of analysis of instability of beams.
- Be able to explain the concepts, quantities and constants that are used in advanced beam calculations

*Skills and abilities*

For a passing grade the student must

- Know how to calculate deformations, stresses and instability load for a straight linear elastic beam with constant or varying arbitrarily shaped cross-section and loaded in 3D by forces, bending moments, torque, secondary moment and eigenstress.
- Know how to calculate, exact or numerically approximately, the stiffness matrix and loading vector for beams of the above kind and how to use these for analysis of structures composed of beams.
- Know how to calculate the cross-section constants for a cross-section of arbitrary shape.
- Know how to make account of a beam design or analysis calculation.
- Know how to use tables and handbooks with information about beam constants and instabilities.

*Judgement and approach*

For a passing grade the student must

- Be able to assess the way of action and properties of a beam (deformation pattern, stiffness properties, stress distribution and instability phenomena) based on the geometrical shape and loading of the beam.
- Be able to assess appropriate method of calculation.

**Contents**

The course relates to methods of calculation för elastic beams with symmetric/unsymetric, open/closed, solid/hollow constant/varying cross sections, exposed to loading in 3D, including distributed bending, torque, secondary moment and eigenstress:

- A summary of different types of beams, phenomena that limit structural serviceability and theories for beam analysis.
- The Bernoulli-Euler and Timoshenko theories for the response to bending moments, shear forces, normal force and eigenstress.
- The St Venants and Vlasov theories for analysis of torsion.
- Second order theory for instability phenomena like buckling in bending and torsion and transverse loading.
- Matrix formulation of beam stiffness and loading for computer based analysis of 3D framework structures.
- Second order theory for analysing instability phenomena such as 3D buckling and tilting.

**Literature**

1) Map with lecture notes and exercises.

2) Austrell. P.-E. et al., CALFEM - A finite element toolbox, The Division of Structural Mechanics, Lund University, distributed by KFS i Lund AB