Designing a curved architectural envelope is a challenging task, as it requires a balance between structural efficiency, fabricability and architectural requirements. In this article, we introduce a new conceptual design tool: Linear-Weingarten membranes, with boundaries either simply supported or realized as funicular cables or arches. We present a generation method for these surfaces, with a focus on how to treat free funicular boundaries. The method uses a surface discretization by triangular meshes, and computes a shape iteratively by a combination of dynamic relaxation and guided projections. The method can be used to generate the shape of self-stressed membranes, funicular vaults, and gridshells. For gridshells, the method allows to find geometries that combine mechanical performance and ease of fabrication.