<p>Analytical solutions to Fluid-Structure Interaction (FSI) problems are almost absent in the literature. However, they are crucial for validation and convergence analysis of numerical methods, as well as for providing insight into the complex coupling dynamics between fluids and solids. In this paper, we derive two analytical and one semi-analytical solutions for three FSI problems, spanning a class of solutions by varying their geometrical and physical parameters. All solutions exhibit complex nonlinear behaviours, which we compare with numerical simulations using a monolithic method. These three FSI problems are described in the cylindrical coordinates, drawing inspiration from Couette flow, with two of them featuring a moving fluid-solid interface and the third incorporating a nonlinear constitutive solid model. To the best of our knowledge, for the first time, we present FSI problems with analytical solutions that include a moving interface.</p>