A comparison of infinite Timoshenko and Euler–Bernoulli beam models on Winkler foundation in the frequency- and time-domain

This paper deals with the dynamic analysis of infinite beam models. The translational and the rotational dynamic stiffness of both Timoshenko and Euler–Bernoulli beams on Winkler foundation are derived and compared in the frequency-domain. The situation of vanishing elastic foundation is included as a special case. Here, special emphasis is placed on the asymptotic behaviour of the derived stiffness expressions for high frequencies, since this is of importance in case of transient excitations. It is shown that the dynamic stiffness of the infinite Timoshenko beam follows a linear function of i ω , whereas rational powers of i ω are involved in case of Euler–Bernoulliʹs model. The stiffness formulations can be transformed into the time-domain using the mixed-variables technique. This is based on a rational approximation of the low-frequency force–displacement relationship and a subsequent algebraic splitting process. At the same time, the high-frequency asymptotic dynamic stiffness is transformed into the time-domain in closed-form. It is shown that the Timoshenko beam is equivalent to a simple dashpot in the high-frequency limit, whereas Euler–Bernoulliʹs beam model leads to fractional derivatives of the unknown state variables in an equivalent time-domain description. This finding confirms the superiority of Timoshenkoʹs model especially for high frequencies and transient excitations. Numerical examples illustrate the differences with respect to the two beam models and demonstrate the applicability of the proposed method for the time-domain transformation of force–displacement relationships.