Abstract This paper considers the computation of reduced-order models for systems of ordinary differential equations that include non-polynomial non-linearities. An targeted example is the case of a geometrically exact model of highly flexible slender structure, that includes, after space discretisation, trigonometric non-linear terms. With a suitable change of variables, this system can be rewritten in an equivalent one with polynomial non-linearities at most quadratic, at the price of introducing additional variables linked to algebraic equations, leading to a differential algebraic set of equations (DAE) to be solved. This DAE is reduced thanks to a normal form parametrisation of its invariant manifolds and selecting a set of master ones. Arbitrary order expansions are detailed for the coefficients of the change of variable and the reduced dynamics, using linear algebra in the space of multivariate polynomials of a given degree. In the case of a single non-linear mode reduction, a criterion to evaluate the quality of the normal form results is also proposed based on an estimation of the convergence radius of the polynomial asymptotic expansion representing truncated series. The method is then applied to compute a single mode reduction of three test cases -- a Duffing oscillator, a simple pendulum and a clamped clamped beam with von~K\'arm\'an model --, in order to investigate the effect of the algebraic part of the DAE on the quality of the model reduction and its validity range. Then, the more involved case of a cantilever beam modelled by geometrically exact finite elements is considered, underlining the ability of the method to produce accurate and converged results in a range of amplitude that can be bounded thanks to a convergence criterion.