The interior problem of an orthotropic strip subject to any given continuous distribution of normal and shear loads is solved by means of a polynomial expansion for the Airy stress function. The polynomial functions defined in the transverse direction are determined recursively by solving a Fredholm equation of second kind. Explicit formulas for displacements are given. A sufficient condition for the convergence of the series expansion is derived. This solution is used to evaluate the error in Timoshenko and higher-order theories. A new beam theory is finally proposed, whose error has the same asymptotic form as second-order theories but approaches zero for strips made of strongly orthotropic material.