Abstract: Transverse vibrations of a beam with uniform inertia and uniform area have been studied extensively by scientists and engineers and closed form solutions leading to simple formulae are readily available for estimating the natural frequencies of a beam when subjected to various types of boundary conditions. However, no such simple formulae exists for beams whose inertia, cross-sectional area and mass vary along the length. In practice, the use of beams with varying sectional properties, such as inertia, area and mass is common, and to estimate the natural frequencies of these beams the engineer has to resort to more expensive methods such as the finite element method. In this paper, a procedure based on the Galerkin method is employed to estimate the first few natural frequencies of a beam with varying sectional properties and including the effects due to rotary inertia. In a particular instance, the numerical values are computed and are verified by using finite element analysis. Also, a formula based on effective inertia and effective area is proposed from which the natural frequencies can be estimated with reasonable accuracy. This formula is simple and is similar to that used for a beam with uniform section properties; also it enables one to evaluate the rotary inertia effect and the shear effect.