The design of curved structural building envelopes is challenging as it requires to account for a wide variety of constraints. In particular, the shape must be mechanically efficient, fabricable, and fit the site geometry. This article shows how a family of surfaces, called isotropic Linear Weingarten surfaces (nicknamed i-liwien), may fulfil all these constraints together, and be used as an intuitive design tool. We start by showing that these shapes are funicular for a uniform vertical load, and that principal stress lines form a conjugate net. This allows in particular for the design of gridshells with planar faces and low bending moments or for the design of self-stressed cable nets cladded by planar glass panels. We then propose a discrete model based on recent advances in discrete differential geometry. We use this model to propose a generation method from boundary curves with two additional control parameters. We demonstrate the shape potential on several examples. The application to shells, membranes, cable-nets and gridshells are discussed.