Lorenz Lachauer

Interactive Vault Design

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.

Three-Dimensional (3D) Equilibrium Analysis of Gothic Masonry Vaults

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.

Geometry of Structural Form

This paper describes a precise geometric method for the inscription of structural constraints into architectural form. Based on techniques from graphic statics, the force distribution in building structures is visualized using geometric diagrams. This diagrammatic representation allows a formal description that shows the relationship between the force flow and the structural form. The formal character of this description enables the direct implementation of a parametric truss geometry that maintains major structural behavioral characteristics under deformation. An interactive model of a structural freeform roof is developed through this link between a parametric truss definition and a design-driving NURBS surface. This allows for an intuitive exploration of the constrained design space in real time. Formal explorations and the comparison with built examples demonstrate the effectiveness of this approach.

Interactive Equilibrium Modelling

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.

Curved Bridge Design

This paper presents a novel approach for the interactive design of linear structures in space. A method is introduced, that provides a maximum of formal freedom in the design of funicular, hence efficient, structures. For a given deck geometry, defined by a design driving NURBS curve via parametric modeling techniques, a tailored relaxation routine allows for controlled, real-time form-finding of the spatial funicular. Subsequently, the equilibrium of the deck is constructed using techniques from graphic statics, combined with a least-square optimization technique. Finally, the method is applied to a design example.

Compression Support Structures for Slabs

This paper presents a novel and rigorous approach for the design of efficient spatial support structures for flat slabs. Based on the given dead load of the slab and the topology of the supporting structure, this method allows form-finding of support structures that are in equilibrium with axial compression forces only. During the form-finding process, the resulting horizontal forces from the compression support structure are balanced through translation and rotation of the slab, and subsequently resolved in the plane of the slab using funicular or trussed strut-and-tie systems. The presented method can be applied for the design of a variety of different typologies of supporting structures, such as inclined columns, curved walls, branching structures and shells. A computational prototype of the approach is implemented as CAD modeling tool, and the potential of the method is shown through two formal design explorations.