### Abstract

This is a kinesthetic activity in which students interact with a physical model to understand the behavior of a cantilever bridge, based on a ‘human cantilever’ demonstration of the Forth Bridge devised by bridge engineer Benjamin Baker.

### Learning Objectives

After this activity, students should be able to:

- Demonstrate how a cantilever bridge relies on a balance of forces
- Describe the role of the different structural elements in a cantilever bridge and identify those that are under compression or under tension
- Reason about the stability of the cantilever and relate changes in one part of the bridge (e.g. greater load) to changes in other parts of the bridge

### Background

Cantilever bridges can be constructed without the need for a large amount of falsework and are therefore useful in spanning difficult crossings. The Forth Bridge is a cantilever railway bridge spanning 8,296 feet over the Firth of Forth in East Scotland, and at the time of its construction in 1890 was the longest-spanning cantilever bridge in the world. This was a bold application of the cantilever principle on such a large scale, at a time when this technique had only been used for major bridges for a few decades.

To demonstrate the stability of the cantilever, Benjamin Baker designed the following demonstration with three men, two chairs, two piles of bricks, and four broomsticks. With their outstretched arms, the people on the left and right serve to transfer the load of the suspended person (center) to the anchors (pile of bricks on left and right).

[shown above] Kaichi Watanabe (center) demonstrating the cantilever principle. (Public Domain)

In this activity, we recreate this classic demonstration, through which students can experience and understand how forces flow through a cantilever bridge, and gain an understanding of the role of balance in this type of structure.

### Materials List

- 4 planks of wood of size 30” x 4” x 1.5”
- 1 plank of wood of size 24” x 6” x 1.5”
- A drill
- Nylon rope (under 0.5” diameter)

### Procedure

Drill 0.5" diameter holes in wood planks and thread rope through the holes as shown in the figure. Ensure that the knots are carefully tied so that they will not come undone under heavy loads. Smoothen the edges of the 30” long cantilever arms so they can be easily gripped.

[shown above] Forth Bridge Model Parts with Distances Marked

[shown above] Length of Anchoring Rope at Far Ends

[shown above] Details of Rope Connections

Introduce the role of balance in a cantilever bridge by comparing the principle of a cantilever bridge to that of a seesaw tied down at one end and supporting a weight at the other end.

Students can engage with this concept through the following think-pair-share polling question:

*What is the minimum force that each anchor needs to withstand, expressed in terms of the suspended weight (W)? Assume that the cantilever arm and the anchor arm are the same length.*

The force exerted on each anchor is:

- Force = W, direction = upwards.
- Force = W, direction = downwards.
- Force = W/2, direction = upwards.
- Force = W/2, direction = downwards

Next, volunteers can participate in forming the demonstration as shown in the figure. Be careful to ensure that the demonstration is stable and balanced, and that the students in the cantilever positions are able to carry the loads.

Students can demonstrate understanding of this exercise by answering questions as follows:

- What do the people standing on the sides feel on their legs? Quantitatively estimate how much force the rope exerts on their feet. Predict how this force change if no one was sitting in the center? What if there were two people sitting in the center?
- Which of the structural elements are in compression (orange) and which are in tension (blue)?

Through this activity, students should recognize that when a load is suspended on the central section of the bridge, the upper beams of the cantilever arms are stretched in tension, while the lower beams are squeezed in compression. Furthermore, they should also recognize that each of the anchors need to support one half of the suspended weight.

### Contributors

This model was assembled by Serguei Bagrianski and Matthew Streeter. Additional teaching materials were developed by Maria Garlock, Negar Khorasani, Aatish Bhatia.

From the lecture: Eiffel's Bridges, Eads Bridge, and Baker's Firth of Forth Bridge