Look at an Umbrella as the Craftsman Looks at It

 

A Craftsman’s Way of Looking at an Umbrella

 

Two photos of an umbrella are shown above, each marginally rotated with respect to the other for the sake of clarity.

Let us assume that the skin of the umbrella was cut from a cloth and stitched continuously on two edges with the ribs, and the third side (the bottom edge) is free. We see eight pieces of cloth, each stitched tightly to the side ribs that go to make the umbrella. Do we learn anything from the figures that gives us anything analytically useful?

The answer starts with looking at the cut piece before it was stitched (in a flat plane). The immediate response, taking mental note that the free surface is curved “inwards” (towards the apex, ‘A’), one may respond with the figure shown here.

But, a structural engineer would think further. The first point (s)he would note is that ‘B-C’ is a curved free-line; by definition, it must be stress-free. Yet, the structural engineer would also “feel” the membrane is “pulling the points ‘B’ and ‘C’ towards the apex, at ‘A’”. It is the writer’s intention to arrive at the explanation not through equations but through visual clues and a logical train of thought.

When an umbrella is opened, the fabric between the ribs is stretched and it is taut when the umbrella is fully open. The membrane between two adjacent ribs is in tension along both the circumferential (θ-direction) and the ‘r’ (radial) direction. This is a trivial observation, as a membrane is uniquely incapable of resisting compression or bending. The above makes it clear that membrane is accommodating the required tension through deformation – in other words, it “yields” to the radial tension, shortening itself in that direction between the nodes. The level of shortening of the radial fibres is not constant between the circumferential nodes ‘B’ and ‘C’. At the nodes, it is minimal and between the ribs, it is significant. It is this differential shortening between two adjacent radial fibres that give rise to the curve of the edge of the fabric between the circumferential nodes, noted in the opening photographs.

Then we conclude the discussion that the craftsman cuts the membrane in a triangle and when stitching it to the ribs, he stretches the skin. This is how the final product, when it is in use, shows a curvature between the circumferential nodes. The craftsman knows his craft, empirically affirmed and now structural engineers are free to analyse the same using mathematics – the two approaches will give precisely the same result. The sketch shows the cloth cut-to-size and ready to be stitched!

Sometimes, students will take in the conceptual explanations more readily than rigorous analysis. Yet, adding such concepts in the syllabus forces the teachers to explain analytically. Students get exacerbated and sigh, “Oh, another equation to remember…”. They learn nothing. The above is a case of something is better than nothing – thinking conceptually.