(Created 2009-08-11.)

BEAM THEORY | VSM091 |

**Aim**

The course shall give knowledge about the action of beams and about theories for calculation of stiffness, deformations, stresses and instability of beams loaded in 3D and with a cross-section of 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 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 loads for a straight linear elastic beam with constant cross-section and loaded in 3D by forces, bending moments, torque and secondary moment.
- know how to calculate the stiffness matrix for beams of the above kind and how to use this matrix 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.

*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.

**Contents**

- A summary of different types of beams, phenomena that limit structural ability and theories for beam analysis.
- The Bernoulli-Euler and Timoshenko theories for the response to bending moments, shear forces and normal force.
- The St Venants and Vlasov theories for analysis of torsion of beams with thick and thin cross section, respectively.
- Matrix formulation of beam stiffness properties 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, 2004, ISBN:91-8855823-1.