The Timoshenko beam theory takes into account the shear deformation and rotational inertia of the beam, while the Euler-Bernoulli beam theory assumes that the beam is long and slender with no shear deformation or rotational inertia.
In-plane beams are those that bend only in a single plane, meaning that the cross-section of the beam remains unchanged during bending. This is often the case for beams with symmetrical cross-sections, such as rectangular or circular beams.
This theory is commonly used in engineering for the analysis of beams subjected to bending and shear, such as in the design of buildings, bridges, and other structures. It is also used in the design of mechanical systems, such as cranes and other lifting equipment.
The main assumptions are that the beam is initially straight, the beam material is homogeneous and isotropic, and that the beam is subjected to small deformations and displacements. Additionally, the theory assumes that the beam is loaded only in the transverse direction and that the cross-section remains unchanged during bending.
This theory is not valid for beams with high shear stresses, thick or non-uniform cross-sections, or beams with large deflections. It also does not take into account the effects of material non-linearity or geometric imperfections, which may be important in some cases.