This paper presents a new computational framework based on Thrust Network Analysis (TNA) for the design of funicular structures. Fast and robust solving algorithms enable the interactive exploration of these constrained structural systems. By giving explicit, bidirectional control over the internal force distribution and overall geometry to the designer, free exploration of these statically highly indeterminate systems is made possible. The equilibrium of funicular compression networks is represented by reciprocal diagrams, which visually express the force dependencies between different parts of the structure. By modifying these diagrams in real-time, the designer is able to explore novel and expressive vaulted geometries that are blurring the difference between shapes associated to typical compression-only forms, obtained e.g. with hanging networks, and freeform surface structures. The power of this framework for design is demonstrated by a user-friendly software implementation, which has been used to design and build a freeform, thin-tile masonry vault.

This study presents a practical method for three-dimensional static equilibrium analysis for masonry vaults using funicular networks. The method, a nonlinear extension of Thrust Network Analysis, is explained, and through three exemplary case studies, the potential of this new research is demonstrated. These examples discuss different assumptions on the "flow of forces" in Gothic quadripartite vaults; visualize the flat-vault equilibrium of rose windows under wind loading; and provide a stability analysis of the intricate nave vaults of Sherborne Abbey, Dorset, England. The presented approach provides insights in structural redundancy of unreinforced masonry structures by quantifying lower bounds on the geometric safety factors. The method for efficient funicular analysis of complex vault geometries furthermore provides the foundation for a fully three-dimensional funicular analysis implementation, extending thrust line analysis to three-dimensional thrust networks, for historic masonry.

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This paper presents a novel method for computer-aided equilibrium modelling of structures in early design stages. Based on the force density method, an iterative procedure is developed that enables the generation of spatial kinematic pin-jointed structures that are in equilibrium close to a given input geometry, while satisfying additional constraints on both geometry and forces. This method forms the core of an interactive form-finding process that consists of alternating steps of modelling and computational optimization. In each modelling step, the user is able to modify geometry, topology, external forces and constraints of the structure. In each optimization step, equilibrium is re-established while respecting the user-defined constraints. A prototype has been implemented within an existing CAD software package, and three examples illustrate the use of the presented method, ranging from a playful exploration of surprising shapes to the rationalization of structural geometry. The method allows to intuitively explore the formal freedom of spatial equilibrium shapes with mixed compression and tension forces, within hard, user-defined constraints. In conclusion, it is claimed that by providing interactive equilibrium modelling methods, the design of new, surprising spatial forms with efficient structural behaviour is facilitated.

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