Maillart truss form-finding

This example shows how to form-find the topology of the roof of the Maillart Chiasso Shed starting from a generic triangulated roof structure

1. Background

Good structural design is achieved when the engineer/architect is able to shape the structure according to its forces ("form follows forces”). The examples below, from the Chiasso Shed, designed in 1924, by Swiss engineer Robert Maillart (1872–1940) is a good example of that:

In the above structure, the inferior members behave as "concrete cable", and were placed along force trajectories, and we see no diagonal members. A structural analysis also revealed that the roof was designed to achieve constant compressive force in the upper part of the roof. Such design is only possible because the geometry of the inferior members is obtained from the loads coming from the roof. The figure below shows the structural concept:

The form of this roof was obtained with graphic statics [1], in this tutorial, we will see how we can start from a typical triangulated roof structure and perform topology changing to achieve the same structural qualities as in Maillart's roof which are:

• a constant force in the upper member, and

• the extinction of the diagonals.

The initial structure is depicted below:

The main features necessary to achieve Maillart's roof example are highlighted below:

2. Initial roof topology

We will achieve so with the help of graphic statics. First, let's see the form and force diagram for the truss in its original configuration. In the toolbar of AGS go to the function Create Form Diagram and select the option FromLines selecting the lines input lines of the structure. Then, assign forces to the 5 loaded edges present in this diagram by clicking on Assign Loads in the toolbar and set the force on the load edges to -10 kN (note the minus sign). Then compute equilibrium and draw the force diagram by clicking in the button: Create Force Diagram. Scale and move the force diagram to the location indicated in the template file and the result should look like the following:

We will highlight in the force diagram the members corresponding to the top chord, and members corresponding to the diagonals. We will act in modifying the force diagram in order to achieve:

• Constant force on the top chord members {19, 22, 9} located in the left slope of the roof and {5, 0, 3} located in the right slope of the roof.

• Removal of diagonals {21, 14, 6, 4} by imposing magnitude of zero to its force, the equivalent of having it collapse to zero length in the force diagram.

The most important segments on the Force diagram and their corresponding duals in the form diagram are illustrated in the following images:

3. Force diagram modification

Here, we want to perform a specific modification on the forces of the structures, and discover which is the shape for the structure that would allow such force distribution. Therefore, we will change the topology of the structure by modifying the force diagram.

Before performing the modification on the force diagram we need to fix the nodes of the form diagram that won't be allowed any movement. Here we will select the top vertices to keep the external load distribution and the constant slope as depicted below:

After the Form Diagram is properly constrained we can perform the movement on the nodes of the structure to achieve the structural qualities mentioned for the Maillart's roof. The following 4 operations will be done using the command Force move nodes that should be used only one time (since it allows for multiple modifications on diagrams and computes equilibrium at the end):

Note that finding the right node to move is sometimes tricky, and for that reason, the edge labels in form and force diagram are turned on during the execution of the command for moving nodes. Once the modifications are done the two diagrams will update. If the result is different from the desired, or you receive an error message, you cal always undo with Ctrl+Z or going to the AGS Menu > Undo

The result is below:

Note that, since forces on diagonals {21, 14, 6, 4} are null, they can be removed from the structure, resulting in a new topology that presents quads instead of triangles.

4. Maillart's topology

This topology now requires only one independent edge, since it is no longer triangulated, now the shape of the structure reacts to the forces applied and the bottom cable is funicular.

Since Maillart's structure is shaped according to the forces applied it is a funicular structure and for that reason, it requires the selection of forces in only one edge.