Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams

The free vibration of functionally graded Timoshenko beams is investigated by developing the dynamic stiffness method. Material properties of the beam are assumed to vary continuously in the thickness direction. The governing differential equations of motion are solved and expressions for axial force, shear force and bending moment are derived. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield the natural frequencies and mode shapes of some illustrative examples. The results are discussed and some conclusions are drawn.