Corentin Fivet

A fully geometric approach for interactive constraint-based structural equilibrium design

This paper introduces computational techniques to support architects and structural designers in the shaping of strut-and-tie networks in static equilibrium. Taking full advantage of geometry, these techniques build on the reciprocal diagrams of graphic statics and enhance the interactive handling of them with two devices: (1) nodes-considered as the only variables-are constrained within Boolean combinations of graphic regions, and (2) the user modifies the diagrams by means of successive operations whose geometric properties do not at any time jeopardize the static equilibrium. This constructive approach enables useful design-oriented capabilities: a graphical control of multiple solutions, the direct switching of the dependencies hierarchy, the execution of dynamic conditional statements using static constraints, the computation of interdependencies, and coordinate-free methods for ensuring consistency between certain continuums of solutions. The paper describes a computer implementation of these capabilities. The initial definition of structural behaviors is of primary importance when seeking material efficiency.A CAD approach is introduced allowing the user to build any plane static equilibrium interactively and graphically.All the design freedoms of the structural problem are permanently contained within dynamic graphical regions of positions.Techniques benefiting from fully geometric abstraction offer some computational simplifications and new capabilities.As a result, this approach frees the designer from the usual hierarchical and chronological structural design processes.

Automatic generation of diverse equilibrium structures through shape grammars and graphic statics

This article presents a computational design methodology that integrates generative (architectural) and analytical (engineering) procedures into a simultaneous design process. By combining shape grammars and graphic statics, the proposed methodology enables the following: (1) rapid generation of diverse, yet statically equilibrated discrete structures; (2) exploration of various design alternatives without any biases toward pre-existing typologies; (3) customization of the framework for unique formulations of design problems and a wide range of applications; and (4) intuitive, bidirectional interaction between the form and forces of the structure through reciprocal diagrams. Design tests presented in this article illustrate the creative potential of the proposed approach and demonstrate the possibility for unbiased explorations of richer and broader design spaces during early stages of design, with much more trial and less error.

Modelling with Forces: Grammar-Based Graphic Statics for Diverse Architectural Structures

Most architectural modelling software provides the user with geometric freedom in absence of performance, while most engineering software mandates pre-determined forms before it can perform any numerical analysis. This trial-and-error process is not only time intensive, but it also hinders free exploration beyond standard designs. This paper proposes a new structural design methodology that integrates the generative (architectural) and the analytical (engineering) procedures into a simultaneous design process, by combining shape grammars and graphic statics. Design tests presented will demonstrate the applicability of this new methodology to various engineering design problems, and demonstrate how the user can explore diverse and unexpected structural alternatives to conventional solutions.

Admissible geometrical domains and graphic statics to evaluate constitutive elements of structural robustness

Structural robustness is related to the insensitivity of a structure to local failure, which is linked to the ability of force redistribution. Current approaches—probabilistic, risk-based, deterministic, and energetic—are poorly suited to inform the design of a structure at conceptual stage. Yet robustness issues must be addressed early in the design process, when interaction with the structural model is still feasible. This article explores how geometrical solution domains provide indicators that summarize the ability of a structure to redistribute load paths. Since geometric methods only build on abstract load path networks, directly connected to form and force diagrams, they suit well for early design explorations. This article presents two case studies, shows how geometrical domains provide qualitative and relevant constitutive indicators of structural robustness, and finally compares the results with indices of deterministic and energetic criteria currently proposed in the literature.

Projective transformations of structural equilibrium

Direct methods that transform the geometry of a structure and conserve its static equilibrium are of particular interest for designers. They allow the exploration of alternative solutions at minimum computational cost. While parallel transformations (stretch, contraction, and skew) are easily understood and have been in use since long, nonparallel transformations that maintain static equilibrium have not been studied in detail. The contributions of this article are as follows. First, an extensive review of transformation methods maintaining static equilibrium of planar and spatial structures is carried out. Second, an alternative construction of projective transformation is developed as an original set of graphical operations. Third, the benefits and limitations of use of projective transformations are discussed for the first time. This article concludes that (1) projective transformations are of practical interest only for a restricted range of structural applications but (2) can significantly simplify the computational problem of geometric exploration of spatial networks in equilibrium with few or no external forces.

Graphic statics and interactive optimization for engineering education

Although conventional structural analysis software is widely used by practicing engineers, its pedagogical value for students is limited, especially in design applications. Nevertheless, there is an established value in students exploring engineering problems through computational means. This paper presents alternative computational techniques and tools that are effective improvements upon structural analysis software in the university classroom. The first set focuses on graphic statics. The second involves interactive evolutionary optimization. The paper provides feedback about their effective implementation in classrooms and demonstrates how the new tools can continue to be used by students beyond the classroom, to expand explorative opportunities for conceptual structural design in practice.

Optimization Formulations for the Design of Low Embodied Energy Structures Made from Reused Elements.

The building sector is one of the major contributors to material resource consumption, greenhouse gas emission and waste production. Load-bearing systems have a particularly large environmental impact because of their material and energy intensive manufacturing process. This paper aims to address the reduction of building structures environmental impacts through reusing structural elements for multiple service lives. Reuse avoids sourcing raw materials and requires little energy for reprocessing. However, to design a new structure reusing elements available from a stock is a challenging problem of combinatorial nature. This is because the structural system layout is a result of the available elements’ mechanical and geometric properties. In this paper, structural optimization formulations are proposed to design truss systems from available stock elements. Minimization of weight, cut-off waste and embodied energy are the objective functions subject to ultimate and serviceability constraints. Case studies focusing on embodied energy minimization are presented for: (1) three roof systems with predefined geometry and topology; (2) a bridge structure whose topology is optimized using the ground structure approach; (3) a geometry optimization to better match the optimal topology from 2 and available stock element lengths. In order to benchmark the energy savings through reuse, the optimal layouts obtained with the proposed methods are compared to weight-optimized solutions made of new material. For these case studies, the methods proposed in this work enable reusing stock elements to design structures embodying up to 71% less energy and hence having a significantly lower environmental impact with respect to structures made of new material.

An Optimized Bracing System for Distributed Lateral Loads

One of the most crucial components of a tall building is its lateral loading system. In this paper, we provide the development of a lateral bracing system that results in bracing material savings of up to 50% relative to a traditional X-Bracing system, as well as lighter corner columns due to the more efficient load paths of the lateral forces to the base. The solution naturally follows a linearized funicular curve, and the result provides a reasonable and replicable system from a manufacturing standpoint.

A geometrical approach to evaluating constitutive elements of structural robustness

The approach proposed here is linked to Maxwell’s reciprocal representation of force and geometry for structural modelling. It is based on the approach of Fivet & Zastavni [2014] of modelling interactive constraint-based structural equilibriums in which geometrical regions are computed to assess a domain of solutions. An examination is undertaken to establish whether the integral of relevant characterising domains can represent an interactive measure of the level of robustness. The approach is applied to case studies, one of which is the Ponte della Musica in Rome, Italy (by the architect Kit Powell-Williams and engineers C. Lotti & Associati and BuroHappold). Structures are analysed in terms of their strength for withstanding different load combinations and degrees of damage. Allowable geometric areas for thrust line(s) are calculated, synthesising the strengths and dimensional constraints, as well as the redistribution of internal forces.