Timoshenko beam theory and Euler-Bernoulli beam theory are two different models used to describe the behavior of beams under various loading conditions. While both theories approximate the behavior of beams, they make different assumptions about the beam's geometry and deformation. Here's a breakdown of the main differences between the two theories:
Assumption about the beam's geometry: In Euler-Bernoulli beam theory, it is assumed that the beam is slender, meaning its cross-sectional dimensions are much smaller compared to its length. This assumption allows for neglecting the effects of shear deformation in the beam. On the other hand, Timoshenko beam theory relaxes this assumption and takes into account the effects of both bending and shear deformation. Timoshenko theory is particularly useful for beams with a low slenderness ratio or when shear deformations are significant.
Deformation model: Euler-Bernoulli beam theory assumes that all sections of the beam remain planar and perpendicular to the beam's neutral axis during deformation. This is known as the "plane sections remain plane" assumption. Consequently, only bending deformation is considered in Euler-Bernoulli theory, while shear deformation is neglected. Timoshenko beam theory, in contrast, allows for both bending and shear deformations. It considers the beam as a combination of bending and shear deformable elements.
Effect of shear deformation: Due to neglecting shear deformation, Euler-Bernoulli beam theory predicts zero shear stress in the beam cross-section. While this approximation is valid for beams with low shear deformations, it becomes inaccurate for beams where shear deformation plays a significant role. Timoshenko beam theory accounts for shear deformation by introducing shear correction factors, leading to more accurate predictions of shear stresses and deformations in the beam.
Natural frequencies: The different deformation models in the two theories also affect the predicted natural frequencies of vibrating beams. Euler-Bernoulli beam theory predicts higher natural frequencies compared to Timoshenko beam theory for a given beam geometry and boundary conditions.
It's worth noting that both Timoshenko beam theory and Euler-Bernoulli beam theory are simplifications of the full three-dimensional elasticity theory for beams. They are useful in engineering and structural analysis as they provide relatively simple mathematical models that capture essential aspects of beam behavior. The choice between the two theories depends on the specific characteristics of the beam, the expected deformations, and the level of accuracy required for the analysis or design.