This chapter considers the first-order shear deformation theory, most commonly known as the Timoshenko beam theory (TBT). The TBT not only brings the transverse shear strain and shear stress into the calculations, but the theory also removes the inconsistency of the classical beam theory, in the sense that the transverse shear force is nonzero even through the constitutive equation by virtue of the transverse shear strain being nonzero. The chapter is concerned with the development of algebraic relationships between the bending solutions for deflections, slopes/rotations, bending moments, and shear forces of the linearized TBT for functionally graded material (FGM) beams in terms of the same quantities of the linearized CBT. To account for the inaccuracy in predicting the transverse shear force magnitude, It introduces shear correction factor. The chapter presents exact solutions to the linear equations of equilibrium of FGM beams without the modified couple stress effect.