Modeling Viscoelastic Foundations for Accurate Structural Analysis: A Review

In practical engineering, foundations often exhibit viscoelastic behavior. Utilizing viscoelastic models to simulate the foundation can more accurately describe its mechanical behavior. Chen and Huang established the stiffness matrix of a Timoshenko beam on a viscoelastic foundation in a moving coordinate system of uniform velocity. They calculated the dynamic response under a harmonic load using the European railway as an example [14, 15]. Kargarnovin and Younesian solved the governing equation using the complex Fourier transform combined with the residual convolution integral theorem. They discussed the response of an infinitely long Timoshenko beam on a viscoelastic foundation under a moving load and a harmonic moving load [16, 17]. Cao et al. studied the response of rigid pavement on a viscoelastic foundation under variable speed load by using an infinite beam. They obtained the dual integral solution by using the double Fourier transform and inverse Fourier transform. They carried out numerical calculations using fast Fourier transform [18]. Sun and Luo studied the response of an Euler-Bernoulli beam on a viscoelastic foundation under a series of moving loads. They proposed an efficient numerical solution method based on fast Fourier transform [19]. Rezvani and Khorramabadi investigated the dynamics of an infinite Timoshenko beam with laminated composite materials on a viscoelastic foundation by using the complex infinite Fourier transform method [20]. Lv et al. regarded asphalt pavement as an infinite beam on a Kelvin viscoelastic foundation. They deduced the analytical solution of its transient equation under a moving load using Green’s function, Laplace transform, and Fourier transform [21]. Basu et al. developed the closed solution of the infinite beam on the Kelvin foundation to the constant velocity concentrated load. They established the dynamic amplification curve of the beam deflection [22]. Ding et al. employed a combination of the Adomian decomposition method, perturbation method, and complex Fourier transform to solve the dynamic response of an infinitely long Timoshenko beam on a nonlinear viscoelastic foundation to a moving load. They considered both the influence of beam shear deformation and foundation shear modulus in their study [23]. Yu et al. deduced the analytical solution subject to arbitrary dynamic load using Fourier and Laplace transform and convolution theorem [24]. Zhao et al. derived the approximate solution of the non-stationary random vibration power spectrum response of an infinite beam on a Kelvin foundation under a moving random load based on the pseudo-excitation method and Fourier transform [25]. Froio et al. studied the relationship between the displacement of the beam and the load velocity of the infinitely uniform beam on the bilinear continuous basis under the constant moving load [26]. Yu and Yuan studied the dynamic response of an infinite beam to a Pasternak foundation under an inclined moving load by using double Fourier transform and inverse transform [27]. Zhen et al. solved the steady-state response of an infinite elastic beam on a nonlinear basis under moving harmonic loads by utilizing Fourier transform, residue theorem, and convolution theorem [28]. Ghannadiasl et al. evaluated the model of an infinite Timoshenko beam subjected to an accelerated moving load on Winkler, Pasternak, and viscoelastic foundation models. They found that the type and depth of the foundation have a significant impact on the dynamic performance of the foundation [29].