In architectural design, geometry is often described using geometric computer-aided design algorithms and sometimes analytical expressions. In contrast, the shape of the Mannheim Multihalle (Mannheim, Germany 1974) and the ICD/ITKE Research Pavilion 2010 (Stuttgart, Germany 2010) derives from the large elastic deformation of flexible elements and is dictated by gravity and material behavior. Recently the word bending active structures was coined to describe these forms.
The trouble with analytical expressions for two-dimensional curves of least bending strain energy. Back in the 19th century the Swiss mathematician Leonard Euler already defined the equilibrium shape of a flexible rod (spline) when bent in two dimensions. Euler studied the buckled pinned strut over a large deflection range, seemingly the first non-linear treatment of elastic instability phenomena. A slender spline is capable of bending far beyond the critical Euler buckling load while remaining in stable equilibrium. When only a pair of balancing forces act at the ends of the initially straight member, the shape of this curve is an elastica. The elastica minimizes bending strain energy. The analytical solution to the shape of the 2D elastica problem, involves the solution of fundamental elliptic integrals and thus in its very nature does not provide a very useful design tool to generate shapes.
The limitations of three-dimensional physical models. The design of the gridshells for the Mannheim Multihalle inverts the geometry of a hanging chain model(in tension) and results in a pure compression shell. For active bending systems, this technique does not necessarily produce the shape as the bending effect of the splines is neglected. But it is a ‘good’ approximated shape. Computationally finding the shape of bending active splines. To relate form to material behavior, non-linear Finite Elements Methods (FEM) have been used that simulate the structural behavior of flexible splines. The simple spline algorithms we have developed, do not employ implicit solution methods like FEM but build on an explicit Dynamic Relaxation technique, initially developed for form-finding of pre-stressed systems. Using compelling case studies of single spline splines and configurations, we explain how our bending formulations generate equilibrium shapes that obey material and statics laws.