Simplified models for determining the response of an isotropic, continuously nonhomogeneous half-plane to a moving distributed line load

The problem of a vertical distributed line load moving with constant speed on the surface of an isotropic half-plane with shear modulus varying continuously with depth has been very recently solved analytically by the authors in an exact manner. The same problem is solved here also analytically under various reasonably simplified assumptions that effectively reduce the coupled system of the two governing partial differential equations of motion into an uncoupled one. The assumptions are zero horizontal displacement, or zero horizontal normal stress, or zero horizontal normal stress plus zero derivative of the horizontal displacement with respect to the vertical coordinate. The method of complex Fourier series involving the horizontal coordinate and the time is used to reduce the uncoupled two partial differential equations to ordinary ones with variable coefficients, which are solved by the method of Frobenius in closed form. Comparison of the approximate solutions corresponding to the aforementioned simplified assumptions against the ‘exact’ solution by means of parametric studies serves to assess the degree of their accuracy for both cases of the shear modulus increasing and decreasing with depth.