A Timoshenko Beam extends the Euler-Bernoulli Beam to incorporate the effects of shear deformation. Shear deformation becomes negligible as a beam’s span:depth ratio increases so can often be ignored.
LUSAS refers to a Timoshenko Beam as a Thick Beam. In SAP2000 all beam elements are Timoshenko beams, although the documentation refers to them as the Bathe and Wilson beam-column formulation, the difference is only in the calculation of the shear correction factor, \(k\).
See Equation 139 of Vibration Problems in Engineering, Timoshenko S. (1937).
\[ EI \frac{\partial^{4} y}{\partial x^{4}} + \frac{\rho A}{g} \frac{\partial^{2}y}{\partial t^{2}} - \Bigg( \frac{\rho I}{g} + \frac{EI \rho}{gkG} \Bigg) \frac{\partial^{4} y}{\partial x^{2} \partial t^{2}} + \frac{\rho I}{g} \frac{\rho}{g k G} \frac{\partial^{4}y}{\partial t^{4}} = 0 \]