Analytical modeling for the design of complex structures

Rasneur, Sylvain [UCL] Zastavni, Denis [UCL]

The observation of complex forms in contemporary architecture reveals a new paradigm : a formal freedom based on non-standard architectural form, whose mathematical formalization is based on NURBS. This new reality fraught with contradictions when we see the complexity of the structures that must support these free surfaces. NURBS were formalized from the work of BEZIER and CASTELJAU, respectively at RENAULT and CITROEN, who developed these parametric polynomial curves to facilitate the design of automotive body parts. A digital curve definition allows it to be drawn by the design office and transmitted to the workshop. But these particular curves do not ensure any mechanical property a priori, they allow above all to communicate digitally. And we inherit today in the design and construction of structures of this geometric freedom. To maintain the freedom of design allowed by digital drawing software from the automotive industry without sacrificing structural efficiency, research studies the relationship between surface geometry, implementation technology, means of assembly and mechanical properties. Shells depend above all on its geometry. It’s not the mass but the shape that determines the strength and statics of a shell element. The presentation proposes to identify two families of main curves of complex shell geometries : geodesic lines and flow lines. These trajectories will be used to design a strut and tie model. Geodesic lines are the curves whose normal vector always corresponds to the normal vector of the surface. Intuitively, it corresponds to the trajectory of an observer who goes straight on the surface. Flow lines are the lines of greatest slope. Inspired by BERANEK’s work on trajectory of shear forces on the fill of bending moments, BORGART proposes the following hypothesis : "The surface of the sum of the curvatures for plates is replaced by the surface of the shell itself. (...) Like a rain flow loads will flow along curves with the steepest ascent on the shell surface to it’s supports. » The flow trajectories then make it possible to visualize how the loads are transmitted from their point of application to the supports. Each part of the shell will dissipate the load through the steepest slope. Until now, the modeling of a strut and tie model on a shell was based on an elastic analysis of the shell considered as a membrane using finite elements and then a connecting strut and tie model was framed on the elastic behavior. However, the steepest slope trajectory geometrically identifies the elastic solution on the shell, with the greatest mechanical stiffness. Therefore, the thesis proposes to use this trajectory as a starting point for modeling struts and tie model of the forces on the shells.