More bridge building
Problem
This problem extends the introductory Bridge Builder problem. You will probably first need to attempt that problem.
Our bridge building engineer wishes to consider the problem of making a longer bridge with $n$ segments along the base.
For a bridge of even length, a single weight 2W is to be hung from the centre. For a bridge of odd length, two weights W will be hung from either side of the central strut, as in the following diagrams:
Part 1) Which lengths of bridge can be created such that there is no net force on any of the pin joints? Try to solve this problem using only vector methods.
Part 2) Which struts do you think will be under most tension/compression. Perform a calculation to see if you are correct.
Part 3) (Very difficult open investigation) Investigate the effects of adding an extra layer of triangles to the top of the bridge or changing the configuration of the supports. Which configurations can you make which can be created with zero net force at all of the joints?
For inspiration in creating these structures you might want to look at the Forth Bridge, in Scotland (see, for example, the Forth Bridge section on the Network Rail website), which is built using the following basic framework structure:
In 1887 the designer of the Forth Bridge, Benjamin Baker, modelled the key principles involved in the bridge's design using two piles of bricks, two ropes, four wooden poles and three people and a seat. The man in the centre is very easily supported as his weight is distibuted throughout the structure.
Extension: The Forth Bridge uses a criss-cross design. Can you work out how the forces are distributed amongst all of its supports? Which parts of the bridge experience the greatest tensions and compressions?
Getting Started
Clearly the bridge will be pulling down on the supports so they will be exerting an upwards reaction force on their supporting pin joints. What does the direction of the reaction force tell us about the forces in the diagaonal support? What does this imply about the direction of the forces in the next strut?
Once you have understood the principles which allow you to determine the locations of tensions and compressions in the bridges you should be able to fill in a picture of tensions and compressions without the need for any numerical calculations.
Student Solutions
Newton, from Macquerie Fields primary school thinks that the bridges of even length will be the most stable and not feel any forces at the joints. Is he right? Can you use vectors and mechanics to help?
Fygliwu presents a very intuitive mechanical explanation:
A bridge of even length should be less stable due to the fact that it has a central point which bears a load. On a bridge of uneven length this load is shared between a set of three points, the two in the middle on the bottom and the one in the middle on the top. This therefore means that the load of an uneven bridge is spread out and therefore dissapated whilst the load of an uneven bridge in focused on a single joint.
Can anyone supply mathematical justification?
Teachers' Resources
Although this problem is quite open ended, the key objective is for students to engage with the distribution of tensions and compressions in a structure as a geometric whole, rather than focussing on the algebra of a typical calculation. Students should learn the power of vector methods to understand the mathematical structure of a problem. Once they develop a feel for the ideas they should be able to create general statements about structures without the need for calculation from the onset. This interplay between geometrical and algebraic arguments is a very important skill for students to begin to develop.
The ideas covered in this problem extend to more challenging investigations of real world structures. Great examples are Forth Bridge in Scotland, the Eiffel Tower and geodesic domes. These can be used to stimulate discussion about the forces in real bridges and structures.
Questions that you may like to pose are: What structural similarities do real world structures have? Why do you think that they share these similarities? Why do you think that the designers chose these structures? How does changing the structure change the location of the greatest tensions and compressions? How do you think the forces are distributed amongst all of these objects? Will there be any net forces at the joints?