This paper develops a one-dimensional extension to classical layered hydraulics that incorporates non-hydrostatic effects. General results for a homogeneous layer in a multi-layer steady flow are applied to single- and two-layer flow over a two-dimensional sill. The equation obtained for single-layer flows is the same as that obtained by Naghdi & Vongsarnpigoon (1986) using the direct theory of constrained fluid sheets, and compares very well with the laboratory measurements of Sivakumaran et al. (1983). The new equation derived for two-layer flows provides excellent agreement with the laboratory measurements of Lawrence (1993). Accurate solutions are obtained for a regime of two-layer flow whose behaviour cannot be explained, even qualitatively, using classical hydraulic theory.