Bridge builder
Problem
A bridge, which balances on two supports and bears a central weight, is to be made in a triangular pattern, as in the diagram. Each segment of the bridge will be either a rope, which must be under tension to be stable, or a spring, which must be under compression to be stable.The pin joints are light and move freely, but will break if subjected to any net force.
In this problem we investigate which parts of the bridge must be ropes and which parts must be springs
Part 1: Imagine that the strut X feels a tension of some unknown magnitude. By considering the vector directions of the forces at the left hand support prove that Y must be in compression if there is to be no net force at the support. Extend this idea to determine which struts in the bridge must be in tension and which struts in the bridge must be in compression if there is to be no net force at
any of the pin joints.
Part 2: Is it possible to use ropes and springs to build a stable bridge of this shape with no net force at any of the pin joints if X is a spring?
For an extension of this problem, why not try the problem More Bridge Building ?
Getting Started
Think about why at least one of the joints necessarily feels a net force for the following diagram:
Once you have understood the principles involved with this piece of the framework you can try gradually to fill in the entire framework with compressions and tensions. Can you do this consistently? If so, what can you say about the net forces at each of the pin joints?
Student Solutions
Andre from Tudor Vianu National College, Bucharest, Romania produced a good solution to this question. Can you build on these ideas to answer the extension parts of the question?
The net force on the horizontal axis must be zero at each pin joint, so the projections of the forces on X and Y on the horizontal axis must have opposite directions. So, even if we do not know their magnitude we can still determine whether they are in compression or in tension.
If X is under tension then we must have:
The forces must be symmetrical in respect to a vertical axis which passes through the central weight. Starting from the left toward the centre, I obtained the following type of forces throughout the framework:
For the second part, a similar analysis shows that the structure must be as follows:
There would be a net downward force at P, so the equilibrium would not be stable.
Teachers' Resources
Why use this problem
Possible Approach
Key Questions
- What are the differences between a spring and a string?
- In order for a pin joint to experience no net forces what must be satisfied?
- How can vectors help us in this problem?
Possible Extension
- The bridge in this question is made from 7 triangular segments. What happens if the weight is hanging from the centre of a bridge with more triangular segments?
- What happens if the bridge is made from isosceles-triangular segments instead of equilateral-triangular segments?
- What would happen if an extra row of triangles were jointed onto the top of the bridge?
- Which springs/ropes would experience the most/least internal forces?