## ABSTRACT

The most commonly used displacement-based theories of beams are: classical beam theory, the first-order shear deformation beam theory, the third-order shear deformation beam theory. This chapter is dedicated to the development of the classical theory of straight beams, accounting for the von Karman nonlinear strain, material gradation through the beam height, and modified couple stress effect. These theories as applied to beams, subjected to external forces that are only functions of x and time t, are governed by differential equations that are function of one spatial coordinate x and time t. The vector approach typically requires identifying an infinitesimal element taken from the continuum with suitable forces displayed on it, while the energy approach requires the construction of the energy functional for the system. The energy approach can be used to derive the governing equations of beams accounting for modified couple stress effects.