Closed form solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width along the beam's length. The solutions are developed taking into account the shear deformation of the beam. The solutions are achieved, in an original way, by transforming the fourth-order differential equations with variable coefficients into fourth-order differential equations with constant coefficients. Though the solutions presented refer to three recurrent variations in the beam cross-section shape, the procedure outlined can be applied to beams with binomial variation (with any exponent) in the depth or width of the cross-section. Moreover, the solutions can be achieved for polynomial, exponential and sinusoidal load conditions. The solutions can be utilized to obtain the stiffness factors and the flexibility coefficients of beams in the analysis of frames. Closed form solutions for longitudinal displacements are also presented. The analytical solutions are applied to four recurrent beams commonly used in civil engineering practice and a comparison with a numerical procedure is made.