Electro-osmosis is an effective way of transporting fluids at smaller scales. Theory for transporting fluids over flat plate is well developed for simple rough surfaces like sinusoidal surfaces are attempted by several researchers. However such surfaces are rare in real life. In this paper, theory is developed for the electroosmotic (EO) flow over rough surfaces with complex shapes, like triangular, trapezoidal, etc. The obtained result was verified with the numerical simulations. Results were obtained using the domain perturbation method for low Reynolds number, constant charge electroosmotic (EO) flow . Results have been obtained up to second-order accuracy of the perturbation parameter. This work will help in designing rough surfaces over a surface, prior to manufacturing, without using simulations, which is the only available method to compute the effect of roughness on flow behavior. The novel concept of the effective shear plane is advanced by this study, which gives insight into the roughness effect on the EO flow.