May 17, 2012, 9:42 pm
Design of a Calatrava-like parametric roof
Firstly, the coordinates of points along the main axis as well as the punctual radii are calculated and saved into a 2-dimmendional array.
In the second step, every two neighboring points are connected by a cylinder with the given radius. In order to prevent sharp steps on changes of radius, a sphere is added at every point and its radius is equal to the greater one of the two neighboring cylinders.
At the same time, there are two triangular polygons created. Two of their points are derived from the corresponding point along the main axis whereas the z-coordinate is increased (or decreased) by the previously calculated punctual radius. The third point is also derived from the point along main axis but the y-coordinate is increased by "size" and the z-coordinate is increased by "slope" * "size" (simplified).
The triangular polygons are a simple solution but they do not render properly in C4D (example: shadows are incorrect). It is better to use extruded volumes with small thickness instead.
Execution of the script in Rhino
Manipulating the parameters
Some renderings of the model in Cinema4D:
The further development includes the implementation of cremona grahic statics to find the funicular form of the main pipe so that only an axial force is present.
Furthermore, a methematical function has been inplemented to find the length of the cantilever arms so that the momentums in each node cancel out. Finally, cover plates with a triangular division have been added.
In order to prevent torsion forces in the main pipe, every node is set to be in equilibrium. The momentums produced by the two opposite cantilever arms have to cancel out. A function has been implemented to calculate the length according to the given momentum and given angle.
M = Axial node momentum
R = Pipe radius
= Cantilever arm thickness
= Material density
A graphic representation of the function (a trace along which the endpoint can be set for a given momentum):
Iterative catenary procedure:
Now, the form of the pipe is not only calculated based on the weights of the cantilevers but also the weight of the pipe itself as well as the cover plates are taken into account. Ofcourse, these are dependent on the form, therefore a formfinding cycle is performed 100 times until a state of conguency is found.
The parametric model has now been used by our group for a project in the structural design class by Professor Schwartz. It has required some adjustments and further development in order to set it into a real scene. The calculation of loads now bases on real material values and considers also the live loads.
In order to crate a convincing model, a function has been implemented to create foundations out of surfaces. Their shape is derived from the shape of the vault whereas the height can be manipulated:
The boundary conditions have been adjusted to the scene the vaultPavillon is set into. The first support is 4.4 meters higher, than the second one. The momentums in the nodes are defined by the following formula:
Because of its large size (pipe length 50m), the glass covers have been replaced by lightweight steel pipes. This solution also reduces the side loads due to wind since it has only a little projected surface. In the future, the steel pipes can be used for ivy growth like in this visualization:
Here is a visualization and plans:
The whole process done by the c# script is described in this project booklet (in German):