The present study proposes a dynamic numerical solution for deflections of curved beam structures. In order to extract characteristic equations of an arch under an in-plane constant moving load, an analysis procedure based on the Euler–Bernoulli beam theory considering polar system is conducted. A prismatic semicircular arch with uniform cross section, in various boundary conditions, is assumed. Radial and tangential displacements, as well as bending moments are obtained using differential quadrature method as a well-known numerical method. In addition to parametric studies, a curved steel bridge as an actual application is analyzed by the mentioned method. By using this differential quadrature technique, the function values and some partial derivatives are approximated by weighting coefficients. Convergence study is carried out to demonstrate the stability of the present method. In order to confirm the high level of accuracy of this approach, some comparisons are made between the results obtained by selected methods such as differential quadrature method, Galerkin method, and finite element method. The results show that in the structural problems with specific geometry, using differential quadrature method, which is independent of domain discretization, is proven to be efficient.