Abstract

This is an in-class activity that uses a piece of paper and candy such as Hershey’s Kisses to demonstrate how curvature can generate stiffness, and to help students understand the important role of shape and form in the stiffness of a structure.

Learning Objectives

After this activity, students should be able to:

• Demonstrate that curvature can increase the stiffness of a structure without adding thickness
• Recognize that there are multiple valid solutions to an engineering design problem
• Relate the forms of certain thin-shell structures to their stiffness

Background

Thin-shell concrete structures are very thin compared to their size, and are typically unsupported by additional structural elements such as columns or buttresses. They can be constructed in the form of flat surfaces, domes, cylindrical sections, may incorporate parts of other curved surfaces such as hyperboloids and hyperbolic paraboloids, or may be inspired by naturalistic forms such as hanging fabric. The form of these concrete shells can combine the inherent strength of curved surfaces with an elegant expression of their thinness. Structural artists who pioneered and built large-scale concrete shells include Eduardo Torroja, Felix Candela, and Heinz Isler.

Eduardo Torroja designed the roof of the Zarzuela Hippodrome in Madrid as a thin-shell horizontal cantilever built out of reinforced concrete.

Roof of the La Zarzuela Racetrack, Madrid, Spain. [Image Credit]: Outisnn / Wikimedia (CC BY-SA 4.0)

The long arm of the cantilevering roof spans 42 feet, and tapers to just a 2 inch thickness at its free end. Torroja achieved such a thin and lightweight roof form by basing the design on a portion of a hyperboloid, a curved surface that has double curvature (curvature in two perpendicular directions).

Modification of original image by William McCallum (CC BY-SA 3.0)

In this activity, students work in groups of two to create a paper cantilever form that can support the greatest possible load. Students are provided with a sheet of letter size paper with a dashed line. When testing the paper cantilever, they are required to hold it from behind this line. The plastic cup is attached with a paper clip to the free end of the cantilever, and is loaded with Hershey’s kisses. A larger number of Hershey’s kisses indicates a stiffer form. Through their explorations, students can discover that forms incorporating curvature have greater stiffness.

Materials List

• For each group: one piece of letter size paper, one plastic cup, one paper clip, about 20 Hershey’s kisses.

Procedure

Draw a dashed line 2 inches from the edge along the long side of the paper to demarcate the support end of the cantilever. Using the paper clip, attach the plastic cup to the opposite end (free end) of the paper.

To help guide students into incorporating curvature into their designs, you can begin by discussing the design of a curved thin-shell structure, such as the roof of Torroja’s Zarzuela racetrack.

Next, ask students to use the piece of paper to develop a structure that can hold as many Hershey’s kisses as possible. Before beginning the activity, students may first predict how many Hershey’s kisses they expect their paper structure to hold.

Students then perform the activity. After sufficient time, have students share their designs with the class, as well as report how many Hershey’s kisses their structures were able to hold. Highlight the different design choices made by the groups whose cantilevers could support the greatest loads. It is often surprising to see how the stiffness of these curved structures exceeded initial expectations.

Finally, have the students summarize what they learnt from this activity with a question about the general conclusions:

The responses to this question as well as the students’ various cantilever designs can be used to assess the learning objectives of this activity.