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Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

Dynamic responses of structures due to earthquake excitation are the important problems in engineering, thus, the information concerned is plenty. However, most of the literature is relating to the discrete methods, particularly to the finite element method (FEM), and the one relating to the method combining both the “continuous” and “discrete” models is rare. The objective of this paper is to provide some information in this respect. First, the analytical solution for the natural frequencies and normal mode shapes of a “continuous” tower, without contacting water (or “dry” tower), carrying an eccentric tip mass possessing rotary inertia is determined. Next, the partial differential equation of motion for the forced vibration of the tower, contacting water (or “wet” tower), subjected to support excitation is transformed into a matrix equation by using the last natural frequencies and normal modes shape of the freely vibrating dry tower. Finally, the numerical integration method is used to solve the matrix equation to yield the seismic response of the wet tower. In theory, the mode superposition method is correct only if the total number of modes considered approaches infinity, however, numerical results of this paper reveal that superposition of only the lowest six modes will yield excellent results to be very close to the corresponding ones obtained from the conventional FEM. For this reason, the CPU time required by the presented approach is less than 5% of that required by the conventional FEM.

In the conventional lumped-mass (model) transfer matrix method (LTMM), one must use different frequency equation and the associated initial parameters to determine the natural frequencies and the corresponding mode shapes of a beam with different boundary (supporting) conditions as one may see from the Appendix at the end of this paper. Besides, the eccentricity of each attached lumped mass is also neglected in the existing LTMM. The purpose of this paper is to present a modified LTMM so that one may easily determine the natural frequencies and the corresponding mode shapes of a multi-step Timoshenko beam with various boundary (supporting) conditions and carrying various concentrated elements with eccentricity of each lumped mass considered, by using the same formulation developed from a beam with “free–free” boundary conditions.To this end, by considering the effect of rotary inertia of each beam segment and the eccentricity of each attached lumped mass, the transfer matrix for a typical station carrying three kinds of concentrated elements is derived. Meanwhile, by considering the shear deformation of each beam segment, the transfer matrix for a typical field is also deduced. It has been found that, by changing the magnitude (from zero to infinity) of each concentrated element, one may easily model many problems studied in the exiting literature concerned. The reliability of the presented approach has been confirmed by the good agreement between the numerical results of this paper and those of the existing literature or the conventional finite element method (FEM).

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.