This study aims to investigate the transient response of homogeneous, isotropic and elastic stepped circular arches under various in-plane dynamic loads in the Laplace domain. The effect of shear deformation is taken into consideration. The governing equations are first obtained in the time domain. Laplace transform is then applied and the obtained canonical form of the first order ordinary differential equations has been solved by the Complementary Functions Method (CFM) in the transformed domain. The fifth-order Runge–Kutta (RK5) method has been applied for the numerical solution of the obtained differential equations. The solutions obtained are transformed to the time domain by an appropriate inverse numerical Laplace transform method. A computer program is coded in Fortran for the forced vibration of the considered structures. Verification and exactness of the written program is performed by comparing the results of the present methods and results of ANSYS which is a commercial finite element program. It is emerged that present method is highly accurate and efficient compared to conventional step by step integration methods.