Denis Zastavni

The structural design of Maillart's Chiasso Shed (1924): A graphic procedure

This paper shows how the geometry of one of Robert Maillarts most intriguing projects, the Chiasso Shed, was created using graphics. Originating in the sixteenth century, graphic vectorial equilibrium has been used for studying a wide variety of structural problems. Among other things, it allows a form to be adjusted so that it only encounters axial loads while guaranteeing equilibrium. Here, we see that the general appearance of the structure was determined with reference to a uniformly distributed load, and then concrete was placed along force trajectories. Maillart allowed for some geometric inaccuracies remaining with regard to the way he designed asymmetric loadings to be supported by stiffening members. In so doing, emphasis is placed on efficient axial forces, similar to our modern strut-and-tie approach, in the design of concrete structures.

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.

A fully graphical approach for limit state analysis of existing structures:Application to plane elastic-plastic bended structures and to plane masonry arches.

ABSTRACT When classical elastic analysis fails to model correctly the structural behavior of historical masonry structures because of the brittle, rigid, anisotropic, and inhomogeneous characteristics of their building material, equilibrium-based limit state analysis constitutes an efficient alternative for their structural assessment. The lack of knowledge about the history of loading makes the actual state of stresses impossible to determine for these statically indeterminate structures. However, Plastic Theory provides a powerful theoretical framework that defines in a rather simple way the structural safety level. The lower-bound theorem of plasticity can be applied using graphic statics because it ensures that equilibrium and yield conditions are respected when applying specific constraints to the nodes of the reciprocal diagrams. This article focuses on limit stat analysis of statically indeterminate structures by means of geometrical considerations using graphic statics reciprocal diagrams. For linear-bended structures, we show that: (1) the conditions of stability can be defined graphically by constructing safety domains; (2) collapse modes can be identified and related to specific reciprocal polygons; and (3) the exact value of the collapse load factor can be deduced graphically from the diagrams. Finally, we extend these results to plane masonry arches in relation with the classical thrust line approach.

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.

Structural Assessment of Masonry Arches Using Admissible Geometrical Domains

Following Mery’s thrust line approach, this paper presents an alternative method to define the structural safety of masonry arches, based on admissible geometrical domains. These are implemented in a parametric model built on the reciprocal diagrams of graphic statics. The application to a case study – a semi-circular masonry arch loaded by a central point load – helps drawing a comparison with the classical geometric safety factor as defined by Jacques Heyman. The model is also used to evaluate the impact of geometrical as well as resistance hypotheses on the structural safety level. Analyses first confirm that stereotomy only slightly influences the load bearing capacity of the arch. They also validate the common use of an infinite compressive strength for arches’ constitutive material, since considering a typical value of 10 MPa reduces structural performances by less than 2%. Finally, a methodology using admissible geometrical domains is suggested to get insights on the robustness of masonry arches.

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.

Maillart’s Design Methods and Sustainable Design

This paper examines the design methods of Robert Maillart (1872-1940), drawing mainly on his well-known Chiasso Shed (1924). It shows that Maillart’s stiffened arch probably could not have been defined through structural analysis alone, which implies that sound structural principles would have had to precede any geometrical definition. His analogical design-based geometry demonstrates a good structural behaviour. Maillart achieved a reliable structure while relying mostly on graphics. It appears that the design maintains the concrete structure under at least partial compression or minimising traction. The authors conclude that preliminary sound structural principles and Maillart’s graphic methods for geometrical definition could help to design a durable and reliable structure with advantages comparable to contemporary goals of sustainable design.