New historical records about the construction of the Arch of Ctesiphon and their impact on the history of structural engineering

A piece of historical research about the construction of the ancient Arch of Taq-iKisra, part of the imperial palace of the Sasanian Empire in the city of Ctesiphon, has been carried out. The information obtained, an analysis using graphic statics, the use of a physical model with hanging chains, and an ad hoc optimization program written in MATLAB have shown that the designer of this sixth century ad arch, a Byzantine named Farghán, was aware of the effects of the uneven distribution of loads and the differential settlements of the foundations on the equilibrium shape of structures working exclusively in compression and was able to control them. This discovery predates the earliest statement about the link between the shape of the catenary and that of an arch, by Robert Hooke, by eleven centuries and makes this building relevant not only because of its historical, archaeological, and architectural importance, but also because of its importance in the history of structural engineering. The building is currently in need of restoration to stop its collapse, and an awareness of the way it was designed could be of practical use for the definition of the intervention needed.

Learning Statics by Visualizing Forces on the Example of a Physical Model of a Truss

The article presents a new didactic tool helping in teaching the structures of students of the Faculty of Architecture. It is an attempt to solve the problem related to the difficulties in teaching structural systems among students of architecture. In the beginning, examples of Graphic-Statics tools supporting an intuitive understanding of the construction work are presented. Then a physical model of the truss was implemented, which responsively presents the values of internal forces using the colors of the luminous bars. The main part of the article describes the design elements of the truss model and presents how it works. Then, the influence of the model on the education of architecture students was checked by means of a questionnaire study. The results showed the great educational usefulness of the proposed solution.

Stability of trusses by graphic statics

This paper presents a graphical method for determining the linearized stiffness and stability of prestressed trusses consisting of rigid bars connected at pinned joints and which possess kinematic freedoms. Key to the construction are the rectangular areas which combine the reciprocal form and force diagrams in the unified Maxwell–Minkowski diagram. The area of each such rectangle is the product of the bar tension and the bar length, and this corresponds to the rotational stiffness of the bar that arises due to the axial force that it carries. The prestress stability of any kinematic freedom may then be assessed using a weighted sum of these areas. The method is generalized to describe the out-of-plane stability of two-dimensional trusses, and to describe three-dimensional trusses in general. The paper also gives a graphical representation of the ‘product forces’ that were introduced by Pellegrino and Calladine to describe the prestress stability of trusses.

Analytical graphic statics

Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived, little emphasis has been placed on studying the mathematics behind these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn a lot by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams, and discontinuous stress functions. This is done by introducing a new, three dimensional force diagram (containing the old two dimensional force diagram) depicting the three dimensional equilibrium of planar force systems. A corresponding three dimensional “form diagram” (dual diagram) is introduced, in which forces are treated as linear functionals (dual vectors). It is shown that the polyhedral stress function introduced by Maxwell is in fact a linear combination of these functionals; and the projective dualities connecting these three dimensional diagrams are also explained.