The theories describing the dynamic behavior of freely vibrating beams and plates could be contained in four main categories as: (i) classical theory, (ii) first order shear deformable theory (FSDT), (iii) higher order shear deformable theories (HSDT) and (iv) three-dimensional theory (3D). Being the beam however the structural element over which the leading modelling idea has originated. In particular, the classical beam theory (i.e. Euler-Bernoulli beam theory) is appreciated by engineers for its own simplicity. However, this theory sometime does not suffice and needs of a step forward which is identified in the use of Timoshenko's theory (1921, Philosophical Magazine, xli, 288-290).