Testing Catenary Arch and Catenary Dome

How do you verify that a shape is a catenary?

A catenary shape is defined as the shape a hanging chain makes, resulting in pure tension. When flipped, the resulting catenary arch should, under equal and opposite gravitational load, be in a state of pure compression.

To verify the catenary shapes that I made in Solidworks, I turned them into a catenary arch and a catenary dome, using the following procedure:

1. Create a sketch of the center of each 'node' in the hanging chain model
2. Copy the sketch into a new Solidworks part file
3. Spline the points together
4. Sweep a cross section along the spline
5. Fix the ends of the arch/corners of the dome
6. Apply gravitational load
7. View results, if the compression load is close to equal throughout the part, then it can be considered a catenary arch/dome

Using the spline took to create a continuous curve and sweeping the cross section along the curve yielded the following parts:

Catenary Arch

Catenary Dome

With the ends of the arch and corners of the dome fixed and gravitational load applied, the stress visualization showed that the parts are indeed approximately catenary shapes (Note that it was easier to flip the gravitational force than it would have been to flip the part):

Catenary Arch Under Graviational Load

Catenary Dome Under Graviational Load

The side view shows that the shapes (except for near the fixtures, especially on the arch) have a consistent stress throughout the part, further proving that they are roughly catenary shapes:

Catenary Arch Under Gravitational Load, Side View

Catenary Dome Under Gravitational Load, Side View

This means that using Solidworks to model a hanging chain could, indeed, be used as a tool for generating catenary curves to be used in gridshell design. Another option I have been pursuing is using Matlab to create a catenary equation for a given size and shape (input desired height and width, and the script solves for the necessary curvature, and outputs the list of points for the desired shape/form)