This study presents a semi-analytical solution with the scaled boundary finite element method (SBFEM) for the sloshing of liquid in vertical cylinders with the coaxial dual circular or arc-shaped porous structures in the context of the linear potential theory. Firstly, the whole flow domain can be divided into three sub-domains with the porous structures and the wall of tank, in which the arc-shaped porous structure should be extended to the whole circle and the perforated coefficient of the extension section is set to infinite. Then the velocity potential of flow in each sub-domain is determined by utilizing the SBFEM for solving the Helmholtz equation or the modified Helmholtz equations, and the fundamental equations and the solution process of the SBFEM for this sloshing problem have been derived in detail. In the SBFEM, one of the superior features is that only the circumferential boundary of the computational domain needs to be discretized with the same form of the finite element. As a result, the spatial dimension of the problem can be reduced by one. As a key point, although there are the dual circular or arc-shaped porous structures, only the wall of tank should be discretized. Meanwhile, the governing equations can be solved analytically in the radial direction by using the Bessel functions or the modified Bessel functions. Numerical examples show that only a small quantity of the discretized elements over the cylindrical boundary can achieve an excellent agreement between the numerical results obtained by the proposed method and the analytical solutions, which demonstrate the accuracy and efficiency of this formulation. Finally, several numerical examples are further studied to investigate the effects of the radius of the porous structures, the porous-effect parameter, the nondimensional wave number, and the opening degree together with the distribution of the centre of the arc-shaped porous structures on the sloshing characteristics of liquid in the cylindrical container with the porous structures.