Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler-Bernoulli beam on elastic foundation

The dynamic response of an infinite Euler-Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.