Timoshenko beam theory

The Timoshenko beam theory was developed by the Ukrainian scientists and mechanics Stepan Tymoshenko at the beginning of the 20th century. As part of the beam theory it explains the oscillatory behavior and the deflection of the clamped beam, which in many parts of classical mechanics, in particular in buildings, bridges or similar is important. In these areas it is important that a bar continues to fulfill its function even when subjected to forces, his behavior has to be as accurate as possible be predicted.

It is an extension of the classical Euler - Bernoulli beam theory, in which additionally modified in addition to the inertia of a deformed beam and shear deformation is taken into account in the equation of motion. Hence, the Bernoulli's assumption that the cross-section of a beam to the beam axis is perpendicular to and after the deformation are no longer fulfilled. By allowing additional (shear ) deformation, the stiffness of the beam is reduced. This has higher deformations and lower natural frequencies of the beam result.

With this addition, the complexity of the underlying mechanical equations increases by an additional spatial derivative of 2nd degree.

Quasi-static Timoshenko beam

In the static Timoshenko beam theory, the deflections of the beam are as

Adopted, the coordinates of a point on the beam, the components of the displacement vector in the three Koordinatenrichtugen, the rotation angle of the normal to the beam axis and the displacement of the beam axis in the direction representing.

The equations of motion are obtained as the following system of uncoupled ordinary differential equations:

The Timoshenko beam theory can be converted into the Euler - Bernoulli beam theory, when the last term is neglected. This is permissible if

Wherein the length of the beam. The Euler - Bernoulli beam theory can thus be viewed as a Spezielfall the Timoshenko beam theory for high shear strength.

Combining the two equations gives the top then a homogeneous beam having a constant cross -section, the following equation of motion:

Credentials

  • Engineering Mechanics
  • Statics
  • Structural Analysis
  • Strength of Materials
  • Construction
  • Engineering Design
Euler–Bernoulli beam theory Girder Jacob Bernoulli
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