The statics of funicular shells is governed by Pucher’s theory. When external loads are known in advance, the horizontal equilibrium of a membrane shell can be fully handled through the Airy stress function and then decoupled from its vertical equilibrium leading to the form-found geometry. Apart from limiting the solution space investigated during the search, the manual prescription of the potential field tends to become cumbersome when either relatively elaborate free-form shell planar footprints are addressed or functional requirements are considered. In this work, a recently formulated isogeometric form-finding strategy is employed to study the effect of different kinematic boundary conditions on the shape of funicular shells made of a unilateral material. The procedure benefits from a nonlinear programming routine to automatically determine a feasible Airy stress function fulfilling concurrent static and functional constraints. Further, use is made of spline technology allowing for smooth surface modelling and enhanced computational efficiency.