An exact dynamic stiffness matrix for a beam is developed by integrating the Rayleigh-Love theory for longitudinal vibration into the Timoshenko theory for bending vibration. In the formulation, the Rayleigh-Love theory accounted for the transverse inertia in longitudinal vibration whereas the Timoshenko beam theory accounted for the effects of shear deformation and rotating inertia in bending vibration. The dynamic stiffness matrix is developed by solving the governing differential equations of motion in free vibration of a Rayleigh-Love bar and a Timoshenko beam and then imposing the boundary conditions for displacements and forces.Next the two dynamic stiffness theories are combined using a unified notation. The ensuing dynamic stiffness matrix is subsequently used for free vibration analysis of uniform and stepped bars as well as frameworks through the application of the Wittrick-Williams algorithm as solution technique. Illustrative examples are given to demonstrate the usefulness of the theory and some of the computed results are compared with published ones. The paper closes with some concluding remarks.

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

The authors are grateful to the reviewers for their careful and studied assessment of the paper. All three reviewers have recommended publication of the paper more or less in its existing form without the need for any major revision. The authors have nevertheless revised the paper by taking into account all of the reviewers' mild and non-insistent comments. The reviewers' comments, subtle though they were, helped the authors to improve the paper. The paper is without doubt much improved as a result of the reviewer's comments. Some details of the revision work are given below. Reviewer 1 The reviewer has made a very important point by implying that the bending moment and shear force at a point on the tapered beam should be given in explicit form as a function of V and x. This is reasonable and justified. The authors have taken this point on board in revising the paper, see Eqs. (10)-(12) of the revised paper. Based on the reviewer's comment the authors also felt that it was instructive to give a new figure showing the sign convention for bending moment and shear force (Fig. 2 of the revised paper). This is necessary to improve the clarity of the paper. Reviewer 2 The reviewer's comment to provide a literature review explaining the contributions made by earlier investigators is perfectly valid and legitimate. The authors have now added an additional paragraph in their introduction, giving a commentary on earlier research. Following the other comment made by the reviewer, a new reference [Ref 21] has been added to the paper. The comments made are all helpful and much appreciated. Reviewer 3 The reviewer has made complimentary remarks in all respect of the authors' work. The authors are deeply grateful for and appreciative of the comments.

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The dynamic stiffness matrix of a coupled axial-bending Timoshenko beam is developed to investigate the free vibration behaviour of such beams and their assemblies. Applying Hamilton's principle, the governing differential equations of motion of a Timoshenko beam in free vibration is derived by considering the axial-bending coupling effect arising from the mass axis eccentricity with the elastic axis of the beam cross-section. The differential equations are then solved in an exact sense, giving expressions for the axial and bending displacements as well as the bending rotation. The expressions for axial force, shear force and bending moment are formed using the natural boundary conditions which resulted from the Hamiltonian formulation. Next, the frequency-dependent dynamic stiffness matrix of the coupled axialbending Timoshenko beam is derived by relating the amplitudes of the axial force, shear force and bending moment to the corresponding amplitudes of axial displacement, bending displacement and bending rotation. The resulting dynamic stiffness matrix is effectively applied to investigate the free vibration behaviour of axial-bending coupled Timoshenko beams by making use of the Wittrick-Williams algorithm as solution technique. The results with emphasis on the axial-bending coupling effects and the importance of the shear deformation and rotatory inertia in free vibration behaviour of coupled axial-bending Timoshenko beams and frameworks are discussed with significant conclusions are drawn.